ffc team mailing list archive

[Mehdi Nikbakht <m.nikbakht@xxxxxxxxxx>]

On Tue, 2010-02-02 at 11:18 +0100, Anders Logg wrote:
> On Tue, Feb 02, 2010 at 11:16:20AM +0100, Mehdi Nikbakht wrote:
> > On Tue, 2010-02-02 at 10:50 +0100, Anders Logg wrote:
> > > On Tue, Feb 02, 2010 at 10:42:23AM +0100, Mehdi Nikbakht wrote:
> > > >
> > > >
> > > > On Tue, 2010-02-02 at 10:30 +0100, Anders Logg wrote:
> > > > > I tried looking at this but I'm unsure how it should be
> > > > > handled. Should a cell_integral class be generated or should a
> > > > > surface_integral class be generated?
> > > > >
> > > >
> > > > We handle terms related to surface integral inside a class derived from
> > > > ufc::cell_integral. I have started working on updating ffcpum module
> > > > which is built against standard ffc.
> > > >
> > > > Mehdi
> > >
> > > So a surface integral should just result in a standard cell integral
> > > being generated? Then what is the point of having *dc? When the code
> > > has been generated, you won't be able to tell which cell integrals
> > > came from *dx and which came from *dc.
> >
> > Although we could have them in a separate class, we handle them inside
> > cell_integral class to have compatibility with ufc interface.
> >
> > Note that having *dc helps us to compute the corresponding terms by
> > using gauss points located on a surface.
> >
> > I don't see the point on being able to tell which cell integrals came
> > from *dx and which one from *dc, we add all of them to the global
> > element tensor.
> 
> My point is that if FFC should treat *dc in exactly the same way as
> *dx, then you might as well just write *dx and we can remove the
> "support" for *dc in FFC.
> 
> Or you could just write dc = dx in your form files.

No, it is not the case. They are handled differently inside ffc, we only
add them inside ufc::cell_integral in the code generation stage.

Mehdi
> 
> --
> Anders
> 
> 
> 
> 
> > Mehdi
> > >
> >
> 
> > // This code conforms with the UFC specification version 1.0
> > // and was automatically generated by FFC version 0.7.0.
> > //
> > // Warning: This code was generated with the option '-l dolfin'
> > // and contains DOLFIN-specific wrappers that depend on DOLFIN.
> >
> > #ifndef __POISSON_H
> > #define __POISSON_H
> >
> > #include <cmath>
> > #include <algorithm>
> > #include <stdexcept>
> > #include <fstream>
> > #include <boost/assign/list_of.hpp>
> > #include <ufc.h>
> > #include <pum/GenericPUM.h>
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_0_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_0_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_0_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_0_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_0_1: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_0_1() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_0_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_0_1();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 6;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 2;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     values[0] = 0;
> >     values[1] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[1] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 2*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[num_derivatives + row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[6][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}, {{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[6][1] = {{1}, {1}, {1}, {1}, {1}, {1}};
> >     static const double D[6][1][2] = {{{1, 0}}, {{1, 0}}, {{1, 0}}, {{0, 1}}, {{0, 1}}, {{0, 1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[2];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 2; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[2] = dof_values[1];
> >     vertex_values[4] = dof_values[2];
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[1] = dof_values[3];
> >     vertex_values[3] = dof_values[4];
> >     vertex_values[5] = dof_values[5];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_finite_element_0_0();
> >       break;
> >     case 1:
> >       return new poisson_0_finite_element_0_1();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_1_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_1_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_1_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_1_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_1_1: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_1_1() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_1_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_1_1();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_1: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_1() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 6;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 2;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     values[0] = 0;
> >     values[1] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[1] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 2*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[num_derivatives + row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[6][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}, {{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[6][1] = {{1}, {1}, {1}, {1}, {1}, {1}};
> >     static const double D[6][1][2] = {{{1, 0}}, {{1, 0}}, {{1, 0}}, {{0, 1}}, {{0, 1}}, {{0, 1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[2];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 2; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[2] = dof_values[1];
> >     vertex_values[4] = dof_values[2];
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[1] = dof_values[3];
> >     vertex_values[3] = dof_values[4];
> >     vertex_values[5] = dof_values[5];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_finite_element_1_0();
> >       break;
> >     case 1:
> >       return new poisson_0_finite_element_1_1();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_2: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_2() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_2()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Discontinuous Lagrange', Cell('triangle', 1, Space(2)), 0)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[1][1] = \
> >     {{1.41421356237309}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[1][1] = \
> >     {{1.41421356237309}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[1][1] = \
> >     {{0}};
> >
> >     static const double dmats1[1][1] = \
> >     {{0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[1][1][2] = {{{0.333333333333333, 0.333333333333333}}};
> >     static const double W[1][1] = {{1}};
> >     static const double D[1][1][1] = {{{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[0];
> >     vertex_values[2] = dof_values[0];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_2();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_0_finite_element_3: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_0_finite_element_3() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_finite_element_3()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_0_finite_element_3();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_0_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_0_0() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_0_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_0_dof_map_0_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_0_1: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_0_1(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_0_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = 0;
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 0;
> >     unsigned int global_offset = 0;
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p0 = boost::assign::list_of(pums[0]);
> >
> >     return new poisson_0_dof_map_0_1(p0);
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_0(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3 + pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3 + pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 4;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];\
> >
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 3;
> >     unsigned int global_offset = m.num_entities[0];
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       dofs[2] = 4;
> >       dofs[3] = 5;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       dofs[2] = 3;
> >       dofs[3] = 5;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       dofs[2] = 3;
> >       dofs[3] = 4;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >     coordinates[3][0] = x[0][0];
> >     coordinates[3][1] = x[0][1];
> >     coordinates[4][0] = x[1][0];
> >     coordinates[4][1] = x[1][1];
> >     coordinates[5][0] = x[2][0];
> >     coordinates[5][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p1 = boost::assign::list_of(pums[0]);
> >
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_dof_map_0_0();
> >       break;
> >     case 1:
> >       return new poisson_0_dof_map_0_1(p1);
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_1_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_1_0() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_1_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_0_dof_map_1_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_1_1: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_1_1(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_1_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = 0;
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 0;
> >     unsigned int global_offset = 0;
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p0 = boost::assign::list_of(pums[0]);
> >
> >     return new poisson_0_dof_map_1_1(p0);
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_1: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_1(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3 + pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3 + pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 4;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];\
> >
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 3;
> >     unsigned int global_offset = m.num_entities[0];
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       dofs[2] = 4;
> >       dofs[3] = 5;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       dofs[2] = 3;
> >       dofs[3] = 5;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       dofs[2] = 3;
> >       dofs[3] = 4;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >     coordinates[3][0] = x[0][0];
> >     coordinates[3][1] = x[0][1];
> >     coordinates[4][0] = x[1][0];
> >     coordinates[4][1] = x[1][1];
> >     coordinates[5][0] = x[2][0];
> >     coordinates[5][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p1 = boost::assign::list_of(pums[0]);
> >
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_dof_map_1_0();
> >       break;
> >     case 1:
> >       return new poisson_0_dof_map_1_1(p1);
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_2: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_2() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_2()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Discontinuous Lagrange', Cell('triangle', 1, Space(2)), 0)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return false;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return true;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[2];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 1;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[2][0];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >
> >       break;
> >     case 1:
> >
> >       break;
> >     case 2:
> >
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = 0.333333333333333*x[0][0] + 0.333333333333333*x[1][0] + 0.333333333333333*x[2][0];
> >     coordinates[0][1] = 0.333333333333333*x[0][1] + 0.333333333333333*x[1][1] + 0.333333333333333*x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_0_dof_map_2();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_0_dof_map_3: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_dof_map_3() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_0_dof_map_3()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_0_dof_map_3();
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the cell
> > /// tensor corresponding to the local contribution to a form from
> > /// the integral over a cell.
> >
> > class poisson_0_cell_integral_0_quadrature: public ufc::cell_integral
> > {
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of tensor for the contribution from a local cell
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = x[1][0] - x[0][0];
> >     const double J_01 = x[2][0] - x[0][0];
> >     const double J_10 = x[1][1] - x[0][1];
> >     const double J_11 = x[2][1] - x[0][1];
> >
> >     // Compute determinant of Jacobian
> >     double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv_00 =  J_11 / detJ;
> >     const double Jinv_01 = -J_01 / detJ;
> >     const double Jinv_10 = -J_10 / detJ;
> >     const double Jinv_11 =  J_00 / detJ;
> >
> >     // Set scale factor
> >     const double det = std::abs(detJ);
> >
> >
> >     // Array of quadrature weights
> >     static const double W1 = 0.5;
> >     // Quadrature points on the UFC reference element: (0.333333333333333, 0.333333333333333)
> >
> >     // Value of basis functions at quadrature points.
> >     static const double FE0[1][3] = \
> >     {{0.333333333333333, 0.333333333333333, 0.333333333333333}};
> >
> >     static const double FE1_C0_D01[1][6] = \
> >     {{-1, 0, 1, 0, 0, 0}};
> >
> >     static const double FE1_C0_D10[1][6] = \
> >     {{-1, 1, 0, 0, 0, 0}};
> >
> >     static const double FE1_C1_D01[1][6] = \
> >     {{0, 0, 0, -1, 0, 1}};
> >
> >     static const double FE1_C1_D10[1][6] = \
> >     {{0, 0, 0, -1, 1, 0}};
> >
> >
> >     // local dimension of the current cell
> >     unsigned int offset = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     offset -= 3;
> >
> >     // Compute element tensor using UFL quadrature representation
> >     // Optimisations: ('simplify expressions', False), ('ignore zero tables', False), ('non zero columns', False), ('remove zero terms', False), ('ignore ones', False)
> >     // Total number of operations to compute element tensor: 1266
> >
> >     // Loop quadrature points for integral
> >     // Number of operations to compute element tensor for following IP loop = 1266
> >     // Only 1 integration point, omitting IP loop.
> >
> >     // Function declarations
> >     double F0 = 0;
> >
> >     // Total number of operations to compute function values = 6
> >     for (unsigned int r = 0; r < 3; r++)
> >     {
> >       F0 += FE0[0][r]*w[1][r];
> >     }// end loop over 'r'
> >     unsigned int m = 0;
> >
> >     // Number of operations for primary indices = 1260
> >     for (unsigned int j = 0; j < 6; j++)
> >     {
> >       for (unsigned int k = 0; k < 6; k++)
> >       {
> >         if ((((0 <= j && j < 3)) && ((0 <= k && k < 3))))
> >         {
> >           // Number of operations to compute entry = 35
> >           A[m] += (((Jinv_00*FE1_C1_D10[0][j] + Jinv_10*FE1_C1_D01[0][j]) + (Jinv_00*FE1_C0_D10[0][j] + Jinv_10*FE1_C0_D01[0][j]))*((Jinv_00*FE1_C1_D10[0][k] + Jinv_10*FE1_C1_D01[0][k]) + (Jinv_00*FE1_C0_D10[0][k] + Jinv_10*FE1_C0_D01[0][k])) + ((Jinv_01*FE1_C0_D10[0][j] + Jinv_11*FE1_C0_D01[0][j]) + (Jinv_01*FE1_C1_D10[0][j] + Jinv_11*FE1_C1_D01[0][j]))*((Jinv_01*FE1_C1_D10[0][k] + Jinv_11*FE1_C1_D01[0][k]) + (Jinv_01*FE1_C0_D10[0][k] + Jinv_11*FE1_C0_D01[0][k])))*F0*W1*det;
> >
> >           ++m;
> >         }
> >       }// end loop over 'k'
> >
> >       // Offset the entries corresponding to enriched terms
> >       if ((((0 <= j && j < 3))))
> >         m += offset;
> >     }// end loop over 'j'
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_cell_integral_0_quadrature(const std::vector<const pum::GenericPUM*>& pums) : ufc::cell_integral(), pums(pums)
> >   {
> >      //Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_cell_integral_0_quadrature()
> >   {
> >      //Do nothing
> >   }
> >
> >
> >   /// Tabulate the tensor for the contribution from a local cell
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c) const
> >   {
> >     // Tabulate regular entires of element tensor
> >     tabulate_regular_tensor(A, w, c);
> >
> >     // local dimension of the current cell
> >     unsigned int num_entries = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     num_entries -= 3;
> >
> >     if (num_entries == 0)
> >       return;
> >
> >
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = x[1][0] - x[0][0];
> >     const double J_01 = x[2][0] - x[0][0];
> >     const double J_10 = x[1][1] - x[0][1];
> >     const double J_11 = x[2][1] - x[0][1];
> >
> >     // Compute determinant of Jacobian
> >     double detJ = J_00*J_11 - J_01*J_10;
> >
> >
> >     // Set scale factor
> >     const double det = std::abs(detJ);
> >
> >     // FIXME: It will crash for multiple discontinuities, if we don't have at least one cell which all dofs are enriched
> >     const unsigned int min_entries = 36;
> >     const unsigned int _num_entries = std::max(num_entries*num_entries, min_entries);
> >
> >     // Resizing and reseting auxiliary tensors
> >     Aa.resize(_num_entries);
> >     std::fill(Aa.begin(), Aa.end(), 0.0);
> >
> >     // Define an array to save current quadrature point
> >     double coordinate[2];
> >
> >     // Define ufc::finite_element object(s) to evalaute shape functions or their derivatives on runtime
> >     poisson_0_finite_element_3  element_0;
> >     poisson_0_finite_element_0  element_1;
> >
> >     // Array of quadrature weights
> >     static const double W1[1] = {0.5};
> >
> >
> >     // Array of quadrature points
> >     static const double P1[2] = \
> >     {0.333333333333333, 0.333333333333333};
> >
> >     // Define vectors for quadrature points and weights(note that the sizes are determined in compile time)
> >     std::vector <double> Wn1;
> >     std::vector <double> Pn1;
> >
> >
> >     // Check whether there is any need to use modified integration scheme
> >     if ((pums[0]->modified_quadrature(c)))
> >     {
> >
> >       const std::vector<double> weight1(W1, W1 + 1);
> >       const std::vector<double> point1(P1, P1 + 2);
> >
> >       ConstQuadratureRule standard_gauss = std::make_pair(point1, weight1);
> >       QuadratureRule modified_gauss;
> >
> >       pums[0]->cell_quadrature_rule(modified_gauss, standard_gauss, c);
> >
> >       Pn1 = modified_gauss.first;
> >       Wn1 = modified_gauss.second;
> >
> >     }
> >     else
> >     {
> >       // Map quadrature points from the reference cell to the physical cell
> >       Wn1.resize(1);;
> >       Pn1.resize(2);;
> >
> >
> >       for (unsigned int i = 0; i < 1; i++)
> >       {
> >         Wn1[i] = W1[i];
> >         for (unsigned int j = 0; j < 2; j++)
> >           Pn1[2*i + j] = x[0][j]*(1.0 - P1[2*i] - P1[2*i + 1]) + x[1][j]*P1[2*i + 1] + x[2][j]*P1[2*i];
> >       }
> >     }
> >
> >
> >     // Return the values of enriched function at the quadrature points
> >     std::vector<double> enriched_values_1;
> >     pums[0]->tabulate_enriched_basis(enriched_values_1, Pn1, c);
> >
> >     // Define auxilary indices: m, n
> >     unsigned int m = 0;
> >     unsigned int n = 0;
> >
> >
> >     // Loop over new quadrature points for integral
> >     for (unsigned int ip = 0; ip < Wn1.size(); ip++)
> >     {
> >       // Evalaute tables and entries in the element tensor, if the enhanced value at this quadrature point is non-zero
> >       if (enriched_values_1[ip] != 0)
> >       {
> >         // Pick up the coordinates of the current quadrature point
> >         coordinate[0] = Pn1[2*ip];
> >         coordinate[1] = Pn1[2*ip + 1];
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)
> >         double value_0[1];
> >         double table_0_D0[3][1];
> >         for (unsigned int j = 0; j < 3; j++)
> >         {
> >           element_0.evaluate_basis(j, value_0, coordinate, c);
> >           for (unsigned int k = 0; k < 1; k++)
> >             table_0_D0[j][k] = value_0[k];
> >         }
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })
> >         double value_1[2];
> >         double table_1_D0[6][2];
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           element_1.evaluate_basis(j, value_1, coordinate, c);
> >           for (unsigned int k = 0; k < 2; k++)
> >             table_1_D0[j][k] = value_1[k];
> >         }
> >
> >
> >         // Creating a table to save the values of derivatives order 1 at the current guass point for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })
> >         double value_2[4];
> >         double table_1_D1[6][4];
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           element_1.evaluate_basis_derivatives(j, 1, value_2, coordinate, c);
> >           for (unsigned int k = 0; k < 4; k++)
> >             table_1_D1[j][k] = value_2[k];
> >         }
> >
> >       // Function declarations
> >       double F0 = 0;
> >
> >       // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += table_0_D0[r][0]*w[1][r];
> >         }// end loop over 'r'
> >
> >       // Number of operations for primary indices = 396
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           for (unsigned int k = 0; k < 6; k++)
> >           {
> >             if (!(((0 <= j && j < 3)) && ((0 <= k && k < 3))))
> >             {
> >               // Move the indices of discontinuous spaces to the end of mixed space
> >               if ((0 <= j && j < 3) && (3 <= k && k < 6))
> >               {
> >                 m = j;
> >                 n = k;
> >               }
> >               else if ((3 <= j && j < 6) && (0 <= k && k < 3))
> >               {
> >                 m = j;
> >                 n = k;
> >               }
> >               else if ((3 <= j && j < 6) && (3 <= k && k < 6))
> >               {
> >                 m = j;
> >                 n = k;
> >               }
> >               // Number of operations to compute entry = 11
> >               Aa[m*6 + n] += ((table_1_D1[j][0] + table_1_D1[j][2])*(table_1_D1[k][2] + table_1_D1[k][0]) + (table_1_D1[j][3] + table_1_D1[j][1])*(table_1_D1[k][3] + table_1_D1[k][1]))*F0*Wn1[ip]*det;
> >             }// end check for enriched entiries
> >           }// end loop over 'k'
> >         }// end loop over 'j'
> >       }
> >     }// end loop over 'ip'
> >
> >     // Check whether the current cell is intersected by discontinuity
> >     if ((pums[0]->modified_quadrature(c)))
> >     {
> >       // Define ufc::finite_element object(s) to evalaute shape functions or their derivatives on runtime
> >       poisson_0_finite_element_2  element_0;
> >       poisson_0_finite_element_0  element_1;
> >
> >     // Array of quadrature weights
> >     static const double W4[2] = {0.5, 0.5};
> >
> >
> >     // Array of quadrature points
> >     static const double P4[2] = \
> >     {0.211324865405187,
> >     0.788675134594813};
> >
> >     // Define vectors for quadrature points and weights
> >     std::vector <double> Wn4(2);
> >     std::vector <double> Pn4(4);
> >
> >
> >       const std::vector<double> weight4(W4, W4 + 2);
> >       const std::vector<double> point4(P4, P4 + 2);
> >
> >       ConstQuadratureRule standard_gauss = std::make_pair(point4, weight4);
> >       QuadratureRule modified_gauss;
> >
> >       pums[0]->surface_quadrature(modified_gauss, standard_gauss, c);
> >
> >       Pn4 = modified_gauss.first;
> >       Wn4 = modified_gauss.second;
> >
> >
> >     // Define and initialize the determinant of Jacobian
> >     const double det = 1.0;
> >
> >         // Loop over new quadrature points for integral
> >       for (unsigned int ip = 0; ip < Wn4.size(); ip++)
> >       {
> >           // Pick up the coordinates of the current quadrature point
> >           coordinate[0] = Pn4[2*ip];
> >           coordinate[1] = Pn4[2*ip + 1];
> >
> >
> >           // Creating a table to save the values of shape functions at the current guass point for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })
> >           double value_0[2];
> >           double table_1_D0[6][2];
> >           for (unsigned int j = 0; j < 6; j++)
> >           {
> >             element_1.evaluate_basis(j, value_0, coordinate, c);
> >             for (unsigned int k = 0; k < 2; k++)
> >               table_1_D0[j][k] = value_0[k];
> >           }
> >
> >
> >           // Creating a table to save the values of shape functions at the current guass point for FiniteElement('Discontinuous Lagrange', Cell('triangle', 1, Space(2)), 0)
> >           double value_1[1];
> >           double table_0_D0[1][1];
> >           for (unsigned int j = 0; j < 1; j++)
> >           {
> >             element_0.evaluate_basis(j, value_1, coordinate, c);
> >             for (unsigned int k = 0; k < 1; k++)
> >               table_0_D0[j][k] = value_1[k];
> >           }
> >
> >         // Number of operations for primary indices = 180
> >           for (unsigned int j = 0; j < 6; j++)
> >           {
> >             for (unsigned int k = 0; k < 6; k++)
> >             {
> >               if (!(((0 <= j && j < 3)) || ((0 <= k && k < 3))))
> >               {
> >                 // Move the indices of discontinuous spaces to the end of mixed space
> >                 if ((3 <= j && j < 6) && (3 <= k && k < 6))
> >                 {
> >                   m = j;
> >                   n = k;
> >                 }
> >                 // Number of operations to compute entry = 5
> >                 Aa[m*6 + n] += table_1_D0[j][1]*table_1_D0[k][1]*w[0][0]*Wn4[ip]*det;
> >               }// end check for enriched entiries
> >             }// end loop over 'k'
> >           }// end loop over 'j'
> >       }// end loop over 'ip'
> >     }
> >
> >
> >     // Pick up entries from the total element tensor for the nodes active in the enrichment
> >
> >     // Determine a vector that contains the local numbering of enriched degrees of freedom in ufc::cell c for the field 0
> >     std::vector<unsigned int> active_dofs_0;
> >     pums[0]->tabulate_enriched_local_dofs(active_dofs_0, c);
> >     std::vector<unsigned int>::iterator it_0_0, it_0_1;
> >
> >
> >     m = 0;
> >     for (unsigned int j = 0; j < 6; j++)
> >       for (unsigned int k = 0; k < 6; k++)
> >         if ((0 <= j && j < 3) && (0 <= k && k < 3))
> >           ++m;
> >         else
> >         {
> >           it_0_0 = find(active_dofs_0.begin(), active_dofs_0.end(), j - 3);
> >           it_0_1 = find(active_dofs_0.begin(), active_dofs_0.end(), k - 3);
> >
> >
> >           // Check whether the entry is coressponding to the active enriched node
> >           if (it_0_0 != active_dofs_0.end() || it_0_1 != active_dofs_0.end())
> >             if (((0 <= j && j < 3)) || ((0 <= k && k < 3)) || (it_0_0 != active_dofs_0.end() && it_0_1 != active_dofs_0.end()))
> >             {
> >               A[m] = Aa[j*6 + k];
> >               ++m;
> >             }
> >         }
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the cell
> > /// tensor corresponding to the local contribution to a form from
> > /// the integral over a cell.
> >
> > class poisson_0_cell_integral_0: public ufc::cell_integral
> > {
> > private:
> >
> >   poisson_0_cell_integral_0_quadrature* integral_0_quadrature;
> >
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of tensor for the contribution from a local cell
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c) const
> >   {
> >     // Do nothing
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_0_cell_integral_0(const std::vector<const pum::GenericPUM*>& pums) : ufc::cell_integral(), pums(pums)
> >   {
> >       integral_0_quadrature = new poisson_0_cell_integral_0_quadrature(pums);
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_0_cell_integral_0()
> >   {
> >       delete integral_0_quadrature;
> >   }
> >
> >
> >   /// Tabulate the tensor for the contribution from a local cell
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c) const
> >   {
> >     const unsigned int num_entries = (3 + pums[0]->enriched_local_dimension(c))*(3 + pums[0]->enriched_local_dimension(c));
> >
> >     for (unsigned int j = 0; j < num_entries; j++)
> >       A[j] = 0;
> >
> >     // Add all contributions to element tensor
> >     integral_0_quadrature->tabulate_tensor(A, w, c);
> >   }
> >
> > };
> >
> > /// This class defines the interface for the assembly of the global
> > /// tensor corresponding to a form with r + n arguments, that is, a
> > /// mapping
> > ///
> > ///     a : V1 x V2 x ... Vr x W1 x W2 x ... x Wn -> R
> > ///
> > /// with arguments v1, v2, ..., vr, w1, w2, ..., wn. The rank r
> > /// global tensor A is defined by
> > ///
> > ///     A = a(V1, V2, ..., Vr, w1, w2, ..., wn),
> > ///
> > /// where each argument Vj represents the application to the
> > /// sequence of basis functions of Vj and w1, w2, ..., wn are given
> > /// fixed functions (coefficients).
> >
> > class poisson_form_0: public ufc::form
> > {
> >   const std::vector<const pum::GenericPUM*>& pums;
> > public:
> >
> >   /// Constructor
> >   poisson_form_0(const std::vector<const pum::GenericPUM*>& pums) : ufc::form(), pums(pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_form_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the form
> >   virtual const char* signature() const
> >   {
> >     return "Form([Integral(Product(Function(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), 1), IndexSum(Product(Indexed(ComponentTensor(Sum(Indexed(SpatialDerivative(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((Index(0),), {Index(0): 2})), MultiIndex((FixedIndex(0),), {})), Indexed(SpatialDerivative(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((Index(0),), {Index(0): 2})), MultiIndex((FixedIndex(1),), {}))), MultiIndex((Index(0),), {Index(0): 2})), MultiIndex((Index(1),), {Index(1): 2})), Indexed(ComponentTensor(Sum(Indexed(SpatialDerivative(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 1), MultiIndex((Index(2),), {Index(2): 2})), MultiIndex((FixedIndex(0),), {})), Indexed(SpatialDerivative(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 1), MultiIndex((Index(2),), {Index(2): 2})), MultiIndex((FixedIndex(1),), {}))), MultiIndex((Index(2),), {Index(2): 2})), MultiIndex((Index(1),), {Index(1): 2}))), MultiIndex((Index(1),), {Index(1): 2}))), Measure('cell', 0, None)), Integral(Product(Constant(Cell('triangle', 1, Space(2)), 0), Product(Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((FixedIndex(1),), {FixedIndex(1): 2})), Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 1), MultiIndex((FixedIndex(1),), {FixedIndex(1): 2})))), Measure('surface', 0, None))])";
> >   }
> >
> >   /// Return the rank of the global tensor (r)
> >   virtual unsigned int rank() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of coefficients (n)
> >   virtual unsigned int num_coefficients() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of cell integrals
> >   virtual unsigned int num_cell_integrals() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the number of exterior facet integrals
> >   virtual unsigned int num_exterior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of interior facet integrals
> >   virtual unsigned int num_interior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new finite element for argument function i
> >   virtual ufc::finite_element* create_finite_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_finite_element_0();
> >       break;
> >     case 1:
> >       return new poisson_0_finite_element_1();
> >       break;
> >     case 2:
> >       return new poisson_0_finite_element_2();
> >       break;
> >     case 3:
> >       return new poisson_0_finite_element_3();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new dof map for argument function i
> >   virtual ufc::dof_map* create_dof_map(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_0_dof_map_0(pums);
> >       break;
> >     case 1:
> >       return new poisson_0_dof_map_1(pums);
> >       break;
> >     case 2:
> >       return new poisson_0_dof_map_2();
> >       break;
> >     case 3:
> >       return new poisson_0_dof_map_3();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new cell integral on sub domain i
> >   virtual ufc::cell_integral* create_cell_integral(unsigned int i) const
> >   {
> >     return new poisson_0_cell_integral_0(pums);
> >   }
> >
> >   /// Create a new exterior facet integral on sub domain i
> >   virtual ufc::exterior_facet_integral* create_exterior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new interior facet integral on sub domain i
> >   virtual ufc::interior_facet_integral* create_interior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_auxiliary_0_finite_element_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_0_finite_element_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_auxiliary_0_finite_element_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_auxiliary_0_finite_element_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_auxiliary_0_dof_map_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_0_dof_map_0() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_auxiliary_0_dof_map_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_auxiliary_0_dof_map_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for the assembly of the global
> > /// tensor corresponding to a form with r + n arguments, that is, a
> > /// mapping
> > ///
> > ///     a : V1 x V2 x ... Vr x W1 x W2 x ... x Wn -> R
> > ///
> > /// with arguments v1, v2, ..., vr, w1, w2, ..., wn. The rank r
> > /// global tensor A is defined by
> > ///
> > ///     A = a(V1, V2, ..., Vr, w1, w2, ..., wn),
> > ///
> > /// where each argument Vj represents the application to the
> > /// sequence of basis functions of Vj and w1, w2, ..., wn are given
> > /// fixed functions (coefficients).
> >
> > class poisson_auxiliary_form_0: public ufc::form
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_form_0() : ufc::form()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_auxiliary_form_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the form
> >   virtual const char* signature() const
> >   {
> >     return "Auxiliary ufc::form to initialize standard functions, apply boundary conditions and obtain ufc::dof_map objects for continuous space(required for PUM objects) for a form containing discontinuous spaces.";
> >   }
> >
> >   /// Return the rank of the global tensor (r)
> >   virtual unsigned int rank() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of coefficients (n)
> >   virtual unsigned int num_coefficients() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of cell integrals
> >   virtual unsigned int num_cell_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of exterior facet integrals
> >   virtual unsigned int num_exterior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of interior facet integrals
> >   virtual unsigned int num_interior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new finite element for argument function i
> >   virtual ufc::finite_element* create_finite_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_auxiliary_0_finite_element_0();
> >       break;
> >     case 1:
> >       return new poisson_auxiliary_0_finite_element_0();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new dof map for argument function i
> >   virtual ufc::dof_map* create_dof_map(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_auxiliary_0_dof_map_0();
> >       break;
> >     case 1:
> >       return new poisson_auxiliary_0_dof_map_0();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new cell integral on sub domain i
> >   virtual ufc::cell_integral* create_cell_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new exterior facet integral on sub domain i
> >   virtual ufc::exterior_facet_integral* create_exterior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new interior facet integral on sub domain i
> >   virtual ufc::interior_facet_integral* create_interior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_1_finite_element_0_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_1_finite_element_0_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_finite_element_0_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_1_finite_element_0_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_1_finite_element_0_1: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_1_finite_element_0_1() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_finite_element_0_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_1_finite_element_0_1();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_1_finite_element_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_1_finite_element_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_finite_element_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 6;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 2;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     values[0] = 0;
> >     values[1] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Extract relevant coefficients
> >       const double coeff0_0 =   coefficients0[dof][0];
> >       const double coeff0_1 =   coefficients0[dof][1];
> >       const double coeff0_2 =   coefficients0[dof][2];
> >
> >       // Compute value(s)
> >       values[1] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >     }
> >
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 2*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     if (0 <= i && i <= 2)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >     if (3 <= i && i <= 5)
> >     {
> >       // Map degree of freedom to element degree of freedom
> >       const unsigned int dof = i - 3;
> >
> >       // Generate scalings
> >       const double scalings_y_0 = 1;
> >       const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >       // Compute psitilde_a
> >       const double psitilde_a_0 = 1;
> >       const double psitilde_a_1 = x;
> >
> >       // Compute psitilde_bs
> >       const double psitilde_bs_0_0 = 1;
> >       const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >       const double psitilde_bs_1_0 = 1;
> >
> >       // Compute basisvalues
> >       const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >       const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >       const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >       // Table(s) of coefficients
> >       static const double coefficients0[3][3] =   \
> >       {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >       {0.471404520791032, 0, 0.333333333333333}};
> >
> >       // Interesting (new) part
> >       // Tables of derivatives of the polynomial base (transpose)
> >       static const double dmats0[3][3] =   \
> >       {{0, 0, 0},
> >       {4.89897948556636, 0, 0},
> >       {0, 0, 0}};
> >
> >       static const double dmats1[3][3] =   \
> >       {{0, 0, 0},
> >       {2.44948974278318, 0, 0},
> >       {4.24264068711928, 0, 0}};
> >
> >       // Compute reference derivatives
> >       // Declare pointer to array of derivatives on FIAT element
> >       double *derivatives = new double [num_derivatives];
> >
> >       // Declare coefficients
> >       double coeff0_0 = 0;
> >       double coeff0_1 = 0;
> >       double coeff0_2 = 0;
> >
> >       // Declare new coefficients
> >       double new_coeff0_0 = 0;
> >       double new_coeff0_1 = 0;
> >       double new_coeff0_2 = 0;
> >
> >       // Loop possible derivatives
> >       for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >       {
> >         // Get values from coefficients array
> >         new_coeff0_0 = coefficients0[dof][0];
> >         new_coeff0_1 = coefficients0[dof][1];
> >         new_coeff0_2 = coefficients0[dof][2];
> >
> >         // Loop derivative order
> >         for (unsigned int j = 0; j < n; j++)
> >         {
> >           // Update old coefficients
> >           coeff0_0 = new_coeff0_0;
> >           coeff0_1 = new_coeff0_1;
> >           coeff0_2 = new_coeff0_2;
> >
> >           if(combinations[deriv_num][j] == 0)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >             new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >             new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >           }
> >           if(combinations[deriv_num][j] == 1)
> >           {
> >             new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >             new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >             new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >           }
> >
> >         }
> >         // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >         derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >       }
> >
> >       // Transform derivatives back to physical element
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         for (unsigned int col = 0; col < num_derivatives; col++)
> >         {
> >           values[num_derivatives + row] += transform[row][col]*derivatives[col];
> >         }
> >       }
> >       // Delete pointer to array of derivatives on FIAT element
> >       delete [] derivatives;
> >
> >       // Delete pointer to array of combinations of derivatives and transform
> >       for (unsigned int row = 0; row < num_derivatives; row++)
> >       {
> >         delete [] combinations[row];
> >         delete [] transform[row];
> >       }
> >
> >       delete [] combinations;
> >       delete [] transform;
> >     }
> >
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[6][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}, {{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[6][1] = {{1}, {1}, {1}, {1}, {1}, {1}};
> >     static const double D[6][1][2] = {{{1, 0}}, {{1, 0}}, {{1, 0}}, {{0, 1}}, {{0, 1}}, {{0, 1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[2];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 2; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[2] = dof_values[1];
> >     vertex_values[4] = dof_values[2];
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[1] = dof_values[3];
> >     vertex_values[3] = dof_values[4];
> >     vertex_values[5] = dof_values[5];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_1_finite_element_0_0();
> >       break;
> >     case 1:
> >       return new poisson_1_finite_element_0_1();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_1_finite_element_1: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_1_finite_element_1() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_finite_element_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_1_finite_element_1();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_1_finite_element_2: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_1_finite_element_2() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_finite_element_2()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_1_finite_element_2();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_1_dof_map_0_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_dof_map_0_0() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_1_dof_map_0_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_1_dof_map_0_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_1_dof_map_0_1: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_dof_map_0_1(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_1_dof_map_0_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = 0;
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 0;
> >     unsigned int global_offset = 0;
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p0 = boost::assign::list_of(pums[0]);
> >
> >     return new poisson_1_dof_map_0_1(p0);
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_1_dof_map_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >   const std::vector<const pum::GenericPUM*>& pums;
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_dof_map_0(const std::vector<const pum::GenericPUM*>& pums) :ufc::dof_map(), pums(pums)
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_1_dof_map_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension + pums[0]->enriched_global_dimension();
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3 + pums[0]->enriched_local_dimension(c);
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3 + pums[0]->enriched_max_local_dimension();
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 4;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];\
> >
> >     // Generate code for tabulating extra degrees of freedom.
> >     unsigned int local_offset = 3;
> >     unsigned int global_offset = m.num_entities[0];
> >
> >     // Calculate local-to-global mapping for the enriched dofs related to the discontinuous field 0
> >     pums[0]->tabulate_enriched_dofs(dofs, c, local_offset, global_offset);
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       dofs[2] = 4;
> >       dofs[3] = 5;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       dofs[2] = 3;
> >       dofs[3] = 5;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       dofs[2] = 3;
> >       dofs[3] = 4;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >     coordinates[3][0] = x[0][0];
> >     coordinates[3][1] = x[0][1];
> >     coordinates[4][0] = x[1][0];
> >     coordinates[4][1] = x[1][1];
> >     coordinates[5][0] = x[2][0];
> >     coordinates[5][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     const std::vector<const pum::GenericPUM*>& p1 = boost::assign::list_of(pums[0]);
> >
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_1_dof_map_0_0();
> >       break;
> >     case 1:
> >       return new poisson_1_dof_map_0_1(p1);
> >       break;
> >     }
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_1_dof_map_1: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_dof_map_1() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_1_dof_map_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_1_dof_map_1();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_1_dof_map_2: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_dof_map_2() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_1_dof_map_2()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_1_dof_map_2();
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the cell
> > /// tensor corresponding to the local contribution to a form from
> > /// the integral over a cell.
> >
> > class poisson_1_cell_integral_0_quadrature: public ufc::cell_integral
> > {
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of tensor for the contribution from a local cell
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = x[1][0] - x[0][0];
> >     const double J_01 = x[2][0] - x[0][0];
> >     const double J_10 = x[1][1] - x[0][1];
> >     const double J_11 = x[2][1] - x[0][1];
> >
> >     // Compute determinant of Jacobian
> >     double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Set scale factor
> >     const double det = std::abs(detJ);
> >
> >
> >     // Array of quadrature weights
> >     static const double W4[4] = {0.159020690871988, 0.0909793091280112, 0.159020690871988, 0.0909793091280112};
> >     // Quadrature points on the UFC reference element: (0.178558728263616, 0.155051025721682), (0.0750311102226081, 0.644948974278318), (0.666390246014701, 0.155051025721682), (0.280019915499074, 0.644948974278318)
> >
> >     // Value of basis functions at quadrature points.
> >     static const double FE0[4][3] = \
> >     {{0.666390246014701, 0.178558728263616, 0.155051025721682},
> >     {0.280019915499074, 0.0750311102226081, 0.644948974278318},
> >     {0.178558728263616, 0.666390246014701, 0.155051025721682},
> >     {0.0750311102226081, 0.280019915499074, 0.644948974278318}};
> >
> >     static const double FE1_C0[4][6] = \
> >     {{0.666390246014701, 0.178558728263616, 0.155051025721682, 0, 0, 0},
> >     {0.280019915499074, 0.0750311102226081, 0.644948974278318, 0, 0, 0},
> >     {0.178558728263616, 0.666390246014701, 0.155051025721682, 0, 0, 0},
> >     {0.0750311102226081, 0.280019915499074, 0.644948974278318, 0, 0, 0}};
> >
> >     static const double FE1_C1[4][6] = \
> >     {{0, 0, 0, 0.666390246014701, 0.178558728263616, 0.155051025721682},
> >     {0, 0, 0, 0.280019915499074, 0.0750311102226081, 0.644948974278318},
> >     {0, 0, 0, 0.178558728263616, 0.666390246014701, 0.155051025721682},
> >     {0, 0, 0, 0.0750311102226081, 0.280019915499074, 0.644948974278318}};
> >
> >
> >     // local dimension of the current cell
> >     unsigned int offset = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     offset -= 3;
> >
> >     // Compute element tensor using UFL quadrature representation
> >     // Optimisations: ('simplify expressions', False), ('ignore zero tables', False), ('non zero columns', False), ('remove zero terms', False), ('ignore ones', False)
> >     // Total number of operations to compute element tensor: 144
> >
> >     // Loop quadrature points for integral
> >     // Number of operations to compute element tensor for following IP loop = 144
> >     for (unsigned int ip = 0; ip < 4; ip++)
> >     {
> >
> >       // Function declarations
> >       double F0 = 0;
> >
> >       // Total number of operations to compute function values = 6
> >       for (unsigned int r = 0; r < 3; r++)
> >       {
> >         F0 += FE0[ip][r]*w[0][r];
> >       }// end loop over 'r'
> >       unsigned int m = 0;
> >
> >       // Number of operations for primary indices = 30
> >       for (unsigned int j = 0; j < 6; j++)
> >       {
> >         if ((((0 <= j && j < 3))))
> >         {
> >           // Number of operations to compute entry = 5
> >           A[m] += (FE1_C0[ip][j] + FE1_C1[ip][j])*F0*W4[ip]*det;
> >
> >           ++m;
> >         }
> >       }// end loop over 'j'
> >     }// end loop over 'ip'
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_cell_integral_0_quadrature(const std::vector<const pum::GenericPUM*>& pums) : ufc::cell_integral(), pums(pums)
> >   {
> >      //Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_cell_integral_0_quadrature()
> >   {
> >      //Do nothing
> >   }
> >
> >
> >   /// Tabulate the tensor for the contribution from a local cell
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c) const
> >   {
> >     // Tabulate regular entires of element tensor
> >     tabulate_regular_tensor(A, w, c);
> >
> >     // local dimension of the current cell
> >     unsigned int num_entries = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     num_entries -= 3;
> >
> >     if (num_entries == 0)
> >       return;
> >
> >
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = x[1][0] - x[0][0];
> >     const double J_01 = x[2][0] - x[0][0];
> >     const double J_10 = x[1][1] - x[0][1];
> >     const double J_11 = x[2][1] - x[0][1];
> >
> >     // Compute determinant of Jacobian
> >     double detJ = J_00*J_11 - J_01*J_10;
> >
> >
> >     // Set scale factor
> >     const double det = std::abs(detJ);
> >
> >     // FIXME: It will crash for multiple discontinuities, if we don't have at least one cell which all dofs are enriched
> >     const unsigned int min_entries = 6;
> >     const unsigned int _num_entries = std::max(num_entries, min_entries);
> >
> >     // Resizing and reseting auxiliary tensors
> >     Aa.resize(_num_entries);
> >     std::fill(Aa.begin(), Aa.end(), 0.0);
> >
> >     // Define an array to save current quadrature point
> >     double coordinate[2];
> >
> >     // Define ufc::finite_element object(s) to evalaute shape functions or their derivatives on runtime
> >     poisson_1_finite_element_1  element_0;
> >     poisson_1_finite_element_0  element_1;
> >
> >     // Array of quadrature weights
> >     static const double W4[4] = {0.159020690871988, 0.0909793091280112, 0.159020690871988, 0.0909793091280112};
> >
> >
> >     // Array of quadrature points
> >     static const double P4[8] = \
> >     {0.178558728263616, 0.155051025721682,
> >     0.0750311102226081, 0.644948974278318,
> >     0.666390246014701, 0.155051025721682,
> >     0.280019915499074, 0.644948974278318};
> >
> >     // Define vectors for quadrature points and weights(note that the sizes are determined in compile time)
> >     std::vector <double> Wn4;
> >     std::vector <double> Pn4;
> >
> >
> >     // Check whether there is any need to use modified integration scheme
> >     if ((pums[0]->modified_quadrature(c)))
> >     {
> >
> >       const std::vector<double> weight4(W4, W4 + 4);
> >       const std::vector<double> point4(P4, P4 + 8);
> >
> >       ConstQuadratureRule standard_gauss = std::make_pair(point4, weight4);
> >       QuadratureRule modified_gauss;
> >
> >       pums[0]->cell_quadrature_rule(modified_gauss, standard_gauss, c);
> >
> >       Pn4 = modified_gauss.first;
> >       Wn4 = modified_gauss.second;
> >
> >     }
> >     else
> >     {
> >       // Map quadrature points from the reference cell to the physical cell
> >       Wn4.resize(4);;
> >       Pn4.resize(8);;
> >
> >
> >       for (unsigned int i = 0; i < 4; i++)
> >       {
> >         Wn4[i] = W4[i];
> >         for (unsigned int j = 0; j < 2; j++)
> >           Pn4[2*i + j] = x[0][j]*(1.0 - P4[2*i] - P4[2*i + 1]) + x[1][j]*P4[2*i + 1] + x[2][j]*P4[2*i];
> >       }
> >     }
> >
> >
> >     // Return the values of enriched function at the quadrature points
> >     std::vector<double> enriched_values_4;
> >     pums[0]->tabulate_enriched_basis(enriched_values_4, Pn4, c);
> >
> >     // Define an auxilary index: m
> >     unsigned int m = 0;
> >
> >
> >     // Loop over new quadrature points for integral
> >     for (unsigned int ip = 0; ip < Wn4.size(); ip++)
> >     {
> >       // Evalaute tables and entries in the element tensor, if the enhanced value at this quadrature point is non-zero
> >       if (enriched_values_4[ip] != 0)
> >       {
> >         // Pick up the coordinates of the current quadrature point
> >         coordinate[0] = Pn4[2*ip];
> >         coordinate[1] = Pn4[2*ip + 1];
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)
> >         double value_0[1];
> >         double table_0_D0[3][1];
> >         for (unsigned int j = 0; j < 3; j++)
> >         {
> >           element_0.evaluate_basis(j, value_0, coordinate, c);
> >           for (unsigned int k = 0; k < 1; k++)
> >             table_0_D0[j][k] = value_0[k];
> >         }
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })
> >         double value_1[2];
> >         double table_1_D0[6][2];
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           element_1.evaluate_basis(j, value_1, coordinate, c);
> >           for (unsigned int k = 0; k < 2; k++)
> >             table_1_D0[j][k] = value_1[k];
> >         }
> >
> >       // Function declarations
> >       double F0 = 0;
> >
> >       // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += table_0_D0[r][0]*w[0][r];
> >         }// end loop over 'r'
> >
> >       // Number of operations for primary indices = 30
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           if (!(((0 <= j && j < 3))))
> >           {
> >             // Move the indices of discontinuous spaces to the end of mixed space
> >             if ((3 <= j && j < 6))
> >             {
> >               m = j;
> >             }
> >             // Number of operations to compute entry = 5
> >             Aa[m] += (table_1_D0[j][0] + table_1_D0[j][1])*F0*Wn4[ip]*det;
> >           }// end check for enriched entiries
> >         }// end loop over 'j'
> >       }
> >     }// end loop over 'ip'
> >
> >
> >     // Pick up entries from the total element tensor for the nodes active in the enrichment
> >
> >     // Determine a vector that contains the local numbering of enriched degrees of freedom in ufc::cell c for the field 0
> >     std::vector<unsigned int> active_dofs_0;
> >     pums[0]->tabulate_enriched_local_dofs(active_dofs_0, c);
> >     std::vector<unsigned int>::iterator it_0_0;
> >
> >
> >     m = 0;
> >     for (unsigned int j = 0; j < 6; j++)
> >       if ((0 <= j && j < 3))
> >         ++m;
> >       else
> >       {
> >         it_0_0 = find(active_dofs_0.begin(), active_dofs_0.end(), j - 3);
> >
> >
> >         // Check whether the entry is coressponding to the active enriched node
> >         if (it_0_0 != active_dofs_0.end())
> >         {
> >           A[m] = Aa[j];
> >           ++m;
> >         }
> >       }
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the cell
> > /// tensor corresponding to the local contribution to a form from
> > /// the integral over a cell.
> >
> > class poisson_1_cell_integral_0: public ufc::cell_integral
> > {
> > private:
> >
> >   poisson_1_cell_integral_0_quadrature* integral_0_quadrature;
> >
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of tensor for the contribution from a local cell
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c) const
> >   {
> >     // Do nothing
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_cell_integral_0(const std::vector<const pum::GenericPUM*>& pums) : ufc::cell_integral(), pums(pums)
> >   {
> >       integral_0_quadrature = new poisson_1_cell_integral_0_quadrature(pums);
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_cell_integral_0()
> >   {
> >       delete integral_0_quadrature;
> >   }
> >
> >
> >   /// Tabulate the tensor for the contribution from a local cell
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c) const
> >   {
> >     const unsigned int num_entries = (3 + pums[0]->enriched_local_dimension(c));
> >
> >     for (unsigned int j = 0; j < num_entries; j++)
> >       A[j] = 0;
> >
> >     // Add all contributions to element tensor
> >     integral_0_quadrature->tabulate_tensor(A, w, c);
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the
> > /// exterior facet tensor corresponding to the local contribution to
> > /// a form from the integral over an exterior facet.
> >
> > class poisson_1_exterior_facet_integral_0_quadrature: public ufc::exterior_facet_integral
> > {
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of the tensor for the contribution from a local exterior facet
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c,
> >                                        unsigned int facet) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >
> >     // Compute determinant of Jacobian
> >
> >     // Compute inverse of Jacobian
> >
> >     // Vertices on edges
> >     static unsigned int edge_vertices[3][2] = {{1, 2}, {0, 2}, {0, 1}};
> >
> >     // Get vertices
> >     const unsigned int v0 = edge_vertices[facet][0];
> >     const unsigned int v1 = edge_vertices[facet][1];
> >
> >     // Compute scale factor (length of edge scaled by length of reference interval)
> >     const double dx0 = x[v1][0] - x[v0][0];
> >     const double dx1 = x[v1][1] - x[v0][1];
> >     const double det = std::sqrt(dx0*dx0 + dx1*dx1);
> >
> >
> >     // Compute facet normals from the facet scale factor constants
> >
> >
> >     // Array of quadrature weights
> >     static const double W2[2] = {0.5, 0.5};
> >     // Quadrature points on the UFC reference element: (0.211324865405187), (0.788675134594813)
> >
> >     // Value of basis functions at quadrature points.
> >     static const double FE0_f0[2][3] = \
> >     {{0, 0.788675134594813, 0.211324865405187},
> >     {0, 0.211324865405187, 0.788675134594813}};
> >
> >     static const double FE0_f1[2][3] = \
> >     {{0.788675134594813, 0, 0.211324865405187},
> >     {0.211324865405187, 0, 0.788675134594813}};
> >
> >     static const double FE0_f2[2][3] = \
> >     {{0.788675134594813, 0.211324865405187, 0},
> >     {0.211324865405187, 0.788675134594813, 0}};
> >
> >     static const double FE1_f0_C0[2][6] = \
> >     {{0, 0.788675134594813, 0.211324865405187, 0, 0, 0},
> >     {0, 0.211324865405187, 0.788675134594813, 0, 0, 0}};
> >
> >     static const double FE1_f0_C1[2][6] = \
> >     {{0, 0, 0, 0, 0.788675134594813, 0.211324865405187},
> >     {0, 0, 0, 0, 0.211324865405187, 0.788675134594813}};
> >
> >     static const double FE1_f1_C0[2][6] = \
> >     {{0.788675134594813, 0, 0.211324865405187, 0, 0, 0},
> >     {0.211324865405187, 0, 0.788675134594813, 0, 0, 0}};
> >
> >     static const double FE1_f1_C1[2][6] = \
> >     {{0, 0, 0, 0.788675134594813, 0, 0.211324865405187},
> >     {0, 0, 0, 0.211324865405187, 0, 0.788675134594813}};
> >
> >     static const double FE1_f2_C0[2][6] = \
> >     {{0.788675134594813, 0.211324865405187, 0, 0, 0, 0},
> >     {0.211324865405187, 0.788675134594813, 0, 0, 0, 0}};
> >
> >     static const double FE1_f2_C1[2][6] = \
> >     {{0, 0, 0, 0.788675134594813, 0.211324865405187, 0},
> >     {0, 0, 0, 0.211324865405187, 0.788675134594813, 0}};
> >
> >
> >     // local dimension of the current cell
> >     unsigned int offset = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     offset -= 3;
> >
> >     // Compute element tensor using UFL quadrature representation
> >     // Optimisations: ('simplify expressions', False), ('ignore zero tables', False), ('non zero columns', False), ('remove zero terms', False), ('ignore ones', False)
> >     switch ( facet )
> >     {
> >     case 0:
> >       {
> >       // Total number of operations to compute element tensor (from this point): 84
> >
> >       // Loop quadrature points for integral
> >       // Number of operations to compute element tensor for following IP loop = 84
> >       for (unsigned int ip = 0; ip < 2; ip++)
> >       {
> >
> >         // Function declarations
> >         double F0 = 0;
> >
> >         // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += FE0_f0[ip][r]*w[1][r];
> >         }// end loop over 'r'
> >         unsigned int m = 0;
> >
> >         // Number of operations for primary indices = 36
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           if ((((0 <= j && j < 3))))
> >           {
> >             // Number of operations to compute entry = 6
> >             A[m] += (FE1_f0_C0[ip][j] + FE1_f0_C1[ip][j])*F0*-1*W2[ip]*det;
> >
> >             ++m;
> >           }
> >         }// end loop over 'j'
> >       }// end loop over 'ip'
> >       }
> >       break;
> >     case 1:
> >       {
> >       // Total number of operations to compute element tensor (from this point): 84
> >
> >       // Loop quadrature points for integral
> >       // Number of operations to compute element tensor for following IP loop = 84
> >       for (unsigned int ip = 0; ip < 2; ip++)
> >       {
> >
> >         // Function declarations
> >         double F0 = 0;
> >
> >         // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += FE0_f1[ip][r]*w[1][r];
> >         }// end loop over 'r'
> >         unsigned int m = 0;
> >
> >         // Number of operations for primary indices = 36
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           if ((((0 <= j && j < 3))))
> >           {
> >             // Number of operations to compute entry = 6
> >             A[m] += (FE1_f1_C0[ip][j] + FE1_f1_C1[ip][j])*F0*-1*W2[ip]*det;
> >
> >             ++m;
> >           }
> >         }// end loop over 'j'
> >       }// end loop over 'ip'
> >       }
> >       break;
> >     case 2:
> >       {
> >       // Total number of operations to compute element tensor (from this point): 84
> >
> >       // Loop quadrature points for integral
> >       // Number of operations to compute element tensor for following IP loop = 84
> >       for (unsigned int ip = 0; ip < 2; ip++)
> >       {
> >
> >         // Function declarations
> >         double F0 = 0;
> >
> >         // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += FE0_f2[ip][r]*w[1][r];
> >         }// end loop over 'r'
> >         unsigned int m = 0;
> >
> >         // Number of operations for primary indices = 36
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           if ((((0 <= j && j < 3))))
> >           {
> >             // Number of operations to compute entry = 6
> >             A[m] += (FE1_f2_C1[ip][j] + FE1_f2_C0[ip][j])*F0*-1*W2[ip]*det;
> >
> >             ++m;
> >           }
> >         }// end loop over 'j'
> >       }// end loop over 'ip'
> >       }
> >       break;
> >     }
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_exterior_facet_integral_0_quadrature(const std::vector<const pum::GenericPUM*>& pums) : ufc::exterior_facet_integral(), pums(pums)
> >   {
> >      //Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_exterior_facet_integral_0_quadrature()
> >   {
> >      //Do nothing
> >   }
> >
> >   /// Tabulate the tensor for the contribution from a local exterior facet
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c,
> >                                unsigned int facet) const
> >   {
> >     // Tabulate regular entires of element tensor
> >     tabulate_regular_tensor(A, w, c, facet);
> >
> >     // local dimension of the current cell
> >     unsigned int num_entries = 3 + pums[0]->enriched_local_dimension(c);
> >
> >
> >     // Remove regular local dimension to obtain number of enriched dofs
> >     num_entries -= 3;
> >
> >     if (num_entries == 0)
> >       return;
> >
> >
> >     // Extract vertex coordinates
> >     const double * const * x = c.coordinates;
> >
> >     // Vertices on edges
> >     static unsigned int edge_vertices[3][2] = {{1, 2}, {0, 2}, {0, 1}};
> >
> >     // Get vertices
> >     const unsigned int v0 = edge_vertices[facet][0];
> >     const unsigned int v1 = edge_vertices[facet][1];
> >
> >     // Compute scale factor (length of edge scaled by length of reference interval)
> >     const double dx0 = x[v1][0] - x[v0][0];
> >     const double dx1 = x[v1][1] - x[v0][1];
> >     const double det = std::sqrt(dx0*dx0 + dx1*dx1);
> >
> >
> >     // Compute facet normals from the facet scale factor constants
> >
> >     // FIXME: It will crash for multiple discontinuities, if we don't have at least one cell which all dofs are enriched
> >     const unsigned int min_entries = 6;
> >     const unsigned int _num_entries = std::max(num_entries, min_entries);
> >
> >     // Resizing and reseting auxiliary tensors
> >     Aa.resize(_num_entries);
> >     std::fill(Aa.begin(), Aa.end(), 0.0);
> >
> >     // Define an array to save current quadrature point
> >     double coordinate[2];
> >
> >     // Define ufc::finite_element object(s) to evalaute shape functions or their derivatives on runtime
> >     poisson_1_finite_element_1  element_0;
> >     poisson_1_finite_element_0  element_1;
> >
> >     // Array of quadrature weights
> >     static const double W2[2] = {0.5, 0.5};
> >
> >
> >     // Array of quadrature points
> >     static const double P2[2] = \
> >     {0.211324865405187,
> >     0.788675134594813};
> >
> >     // Define vectors for quadrature points and weights(note that the sizes are determined in compile time)
> >     std::vector <double> Wn2;
> >     std::vector <double> Pn2;
> >
> >
> >     // Check whether there is any need to use modified integration scheme
> >     if ((pums[0]->modified_quadrature(c, facet)))
> >     {
> >
> >       const std::vector<double> weight2(W2, W2 + 2);
> >       const std::vector<double> point2(P2, P2 + 2);
> >
> >       ConstQuadratureRule standard_gauss = std::make_pair(point2, weight2);
> >       QuadratureRule modified_gauss;
> >
> >       pums[0]->facet_quadrature_rule(modified_gauss, standard_gauss, c, facet);
> >
> >       Pn2 = modified_gauss.first;
> >       Wn2 = modified_gauss.second;
> >
> >     }
> >     else
> >     {
> >       // Map quadrature points from the reference cell to the physical cell
> >       Wn2.resize(2);;
> >       Pn2.resize(4);;
> >
> >
> >       for (unsigned int i = 0; i < 2; i++)
> >       {
> >         Wn2[i] = W2[i];
> >         for (unsigned int j = 0; j < 2; j++)
> >           Pn2[2*i + j] = x[v0][j]*(1.0 - P2[i]) + x[v1][j]*P2[i];
> >       }
> >     }
> >
> >
> >     // Return the values of enriched function at the quadrature points
> >     std::vector<double> enriched_values_2;
> >     pums[0]->tabulate_enriched_basis(enriched_values_2, Pn2, c);
> >
> >     // Define an auxilary index: m
> >     unsigned int m = 0;
> >
> >
> >     // Loop over new quadrature points for integral
> >     for (unsigned int ip = 0; ip < Wn2.size(); ip++)
> >     {
> >       // Evalaute tables and entries in the element tensor, if the enhanced value at this quadrature point is non-zero
> >       if (enriched_values_2[ip] != 0)
> >       {
> >         // Pick up the coordinates of the current quadrature point
> >         coordinate[0] = Pn2[2*ip];
> >         coordinate[1] = Pn2[2*ip + 1];
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)
> >         double value_0[1];
> >         double table_0_D0[3][1];
> >         for (unsigned int j = 0; j < 3; j++)
> >         {
> >           element_0.evaluate_basis(j, value_0, coordinate, c);
> >           for (unsigned int k = 0; k < 1; k++)
> >             table_0_D0[j][k] = value_0[k];
> >         }
> >
> >
> >         // Creating a table to save the values of shape functions at the current guass point for MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })
> >         double value_1[2];
> >         double table_1_D0[6][2];
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           element_1.evaluate_basis(j, value_1, coordinate, c);
> >           for (unsigned int k = 0; k < 2; k++)
> >             table_1_D0[j][k] = value_1[k];
> >         }
> >
> >       // Function declarations
> >       double F0 = 0;
> >
> >       // Total number of operations to compute function values = 6
> >         for (unsigned int r = 0; r < 3; r++)
> >         {
> >           F0 += table_0_D0[r][0]*w[1][r];
> >         }// end loop over 'r'
> >
> >       // Number of operations for primary indices = 36
> >         for (unsigned int j = 0; j < 6; j++)
> >         {
> >           if (!(((0 <= j && j < 3))))
> >           {
> >             // Move the indices of discontinuous spaces to the end of mixed space
> >             if ((3 <= j && j < 6))
> >             {
> >               m = j;
> >             }
> >             // Number of operations to compute entry = 6
> >             Aa[m] += (table_1_D0[j][0] + table_1_D0[j][1])*F0*-1*Wn2[ip]*det;
> >           }// end check for enriched entiries
> >         }// end loop over 'j'
> >       }
> >     }// end loop over 'ip'
> >
> >
> >     // Pick up entries from the total element tensor for the nodes active in the enrichment
> >
> >     // Determine a vector that contains the local numbering of enriched degrees of freedom in ufc::cell c for the field 0
> >     std::vector<unsigned int> active_dofs_0;
> >     pums[0]->tabulate_enriched_local_dofs(active_dofs_0, c);
> >     std::vector<unsigned int>::iterator it_0_0;
> >
> >
> >     m = 0;
> >     for (unsigned int j = 0; j < 6; j++)
> >       if ((0 <= j && j < 3))
> >         ++m;
> >       else
> >       {
> >         it_0_0 = find(active_dofs_0.begin(), active_dofs_0.end(), j - 3);
> >
> >
> >         // Check whether the entry is coressponding to the active enriched node
> >         if (it_0_0 != active_dofs_0.end())
> >         {
> >           A[m] = Aa[j];
> >           ++m;
> >         }
> >       }
> >   }
> >
> > };
> >
> > /// This class defines the interface for the tabulation of the
> > /// exterior facet tensor corresponding to the local contribution to
> > /// a form from the integral over an exterior facet.
> >
> > class poisson_1_exterior_facet_integral_0: public ufc::exterior_facet_integral
> > {
> > private:
> >
> >   poisson_1_exterior_facet_integral_0_quadrature* integral_0_quadrature;
> >
> >
> >   const std::vector<const pum::GenericPUM*>& pums;
> >   mutable std::vector <double> Aa;
> >   mutable std::vector <double> Af;
> >
> >
> >
> >   /// Tabulate the regular entities of the tensor for the contribution from a local exterior facet
> >   virtual void tabulate_regular_tensor(double* A,
> >                                        const double * const * w,
> >                                        const ufc::cell& c,
> >                                        unsigned int facet) const
> >   {
> >     // Do nothing
> >   }
> >
> > public:
> >
> >   /// Constructor
> >   poisson_1_exterior_facet_integral_0(const std::vector<const pum::GenericPUM*>& pums) : ufc::exterior_facet_integral(), pums(pums)
> >   {
> >       integral_0_quadrature = new poisson_1_exterior_facet_integral_0_quadrature(pums);
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_1_exterior_facet_integral_0()
> >   {
> >       delete integral_0_quadrature;
> >   }
> >
> >   /// Tabulate the tensor for the contribution from a local exterior facet
> >   virtual void tabulate_tensor(double* A,
> >                                const double * const * w,
> >                                const ufc::cell& c,
> >                                unsigned int facet) const
> >   {
> >     const unsigned int num_entries = (3 + pums[0]->enriched_local_dimension(c));
> >
> >     for (unsigned int j = 0; j < num_entries; j++)
> >       A[j] = 0;
> >
> >     // Add all contributions to element tensor
> >     integral_0_quadrature->tabulate_tensor(A, w, c, facet);
> >   }
> >
> > };
> >
> > /// This class defines the interface for the assembly of the global
> > /// tensor corresponding to a form with r + n arguments, that is, a
> > /// mapping
> > ///
> > ///     a : V1 x V2 x ... Vr x W1 x W2 x ... x Wn -> R
> > ///
> > /// with arguments v1, v2, ..., vr, w1, w2, ..., wn. The rank r
> > /// global tensor A is defined by
> > ///
> > ///     A = a(V1, V2, ..., Vr, w1, w2, ..., wn),
> > ///
> > /// where each argument Vj represents the application to the
> > /// sequence of basis functions of Vj and w1, w2, ..., wn are given
> > /// fixed functions (coefficients).
> >
> > class poisson_form_1: public ufc::form
> > {
> >   const std::vector<const pum::GenericPUM*>& pums;
> > public:
> >
> >   /// Constructor
> >   poisson_form_1(const std::vector<const pum::GenericPUM*>& pums) : ufc::form(), pums(pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_form_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the form
> >   virtual const char* signature() const
> >   {
> >     return "Form([Integral(Product(Function(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), 0), Sum(Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((FixedIndex(0),), {FixedIndex(0): 2})), Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((FixedIndex(1),), {FixedIndex(1): 2})))), Measure('cell', 0, None)), Integral(Product(IntValue(-1, (), (), {}), Product(Function(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), 1), Sum(Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((FixedIndex(0),), {FixedIndex(0): 2})), Indexed(BasisFunction(MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) }), 0), MultiIndex((FixedIndex(1),), {FixedIndex(1): 2}))))), Measure('exterior_facet', 0, None))])";
> >   }
> >
> >   /// Return the rank of the global tensor (r)
> >   virtual unsigned int rank() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the number of coefficients (n)
> >   virtual unsigned int num_coefficients() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of cell integrals
> >   virtual unsigned int num_cell_integrals() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the number of exterior facet integrals
> >   virtual unsigned int num_exterior_facet_integrals() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the number of interior facet integrals
> >   virtual unsigned int num_interior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new finite element for argument function i
> >   virtual ufc::finite_element* create_finite_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_1_finite_element_0();
> >       break;
> >     case 1:
> >       return new poisson_1_finite_element_1();
> >       break;
> >     case 2:
> >       return new poisson_1_finite_element_2();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new dof map for argument function i
> >   virtual ufc::dof_map* create_dof_map(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_1_dof_map_0(pums);
> >       break;
> >     case 1:
> >       return new poisson_1_dof_map_1();
> >       break;
> >     case 2:
> >       return new poisson_1_dof_map_2();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new cell integral on sub domain i
> >   virtual ufc::cell_integral* create_cell_integral(unsigned int i) const
> >   {
> >     return new poisson_1_cell_integral_0(pums);
> >   }
> >
> >   /// Create a new exterior facet integral on sub domain i
> >   virtual ufc::exterior_facet_integral* create_exterior_facet_integral(unsigned int i) const
> >   {
> >     return new poisson_1_exterior_facet_integral_0(pums);
> >   }
> >
> >   /// Create a new interior facet integral on sub domain i
> >   virtual ufc::interior_facet_integral* create_interior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for a finite element.
> >
> > class poisson_auxiliary_1_finite_element_0: public ufc::finite_element
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_1_finite_element_0() : ufc::finite_element()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_auxiliary_1_finite_element_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the finite element
> >   virtual const char* signature() const
> >   {
> >     return "FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return the cell shape
> >   virtual ufc::shape cell_shape() const
> >   {
> >     return ufc::triangle;
> >   }
> >
> >   /// Return the dimension of the finite element function space
> >   virtual unsigned int space_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the rank of the value space
> >   virtual unsigned int value_rank() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the dimension of the value space for axis i
> >   virtual unsigned int value_dimension(unsigned int i) const
> >   {
> >     return 1;
> >   }
> >
> >   /// Evaluate basis function i at given point in cell
> >   virtual void evaluate_basis(unsigned int i,
> >                               double* values,
> >                               const double* coordinates,
> >                               const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Reset values
> >     *values = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Extract relevant coefficients
> >     const double coeff0_0 = coefficients0[dof][0];
> >     const double coeff0_1 = coefficients0[dof][1];
> >     const double coeff0_2 = coefficients0[dof][2];
> >
> >     // Compute value(s)
> >     *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
> >   }
> >
> >   /// Evaluate all basis functions at given point in cell
> >   virtual void evaluate_basis_all(double* values,
> >                                   const double* coordinates,
> >                                   const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis() is not yet implemented.");
> >   }
> >
> >   /// Evaluate order n derivatives of basis function i at given point in cell
> >   virtual void evaluate_basis_derivatives(unsigned int i,
> >                                           unsigned int n,
> >                                           double* values,
> >                                           const double* coordinates,
> >                                           const ufc::cell& c) const
> >   {
> >     // Extract vertex coordinates
> >     const double * const * element_coordinates = c.coordinates;
> >
> >     // Compute Jacobian of affine map from reference cell
> >     const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
> >     const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
> >     const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
> >     const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
> >
> >     // Compute determinant of Jacobian
> >     const double detJ = J_00*J_11 - J_01*J_10;
> >
> >     // Compute inverse of Jacobian
> >
> >     // Get coordinates and map to the reference (UFC) element
> >     double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
> >                 element_coordinates[0][0]*element_coordinates[2][1] +\
> >                 J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
> >     double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
> >                 element_coordinates[1][0]*element_coordinates[0][1] -\
> >                 J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
> >
> >     // Map coordinates to the reference square
> >     if (std::abs(y - 1.0) < 1e-14)
> >       x = -1.0;
> >     else
> >       x = 2.0 *x/(1.0 - y) - 1.0;
> >     y = 2.0*y - 1.0;
> >
> >     // Compute number of derivatives
> >     unsigned int num_derivatives = 1;
> >
> >     for (unsigned int j = 0; j < n; j++)
> >       num_derivatives *= 2;
> >
> >
> >     // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
> >     unsigned int **combinations = new unsigned int *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       combinations[j] = new unsigned int [n];
> >       for (unsigned int k = 0; k < n; k++)
> >         combinations[j][k] = 0;
> >     }
> >
> >     // Generate combinations of derivatives
> >     for (unsigned int row = 1; row < num_derivatives; row++)
> >     {
> >       for (unsigned int num = 0; num < row; num++)
> >       {
> >         for (unsigned int col = n-1; col+1 > 0; col--)
> >         {
> >           if (combinations[row][col] + 1 > 1)
> >             combinations[row][col] = 0;
> >           else
> >           {
> >             combinations[row][col] += 1;
> >             break;
> >           }
> >         }
> >       }
> >     }
> >
> >     // Compute inverse of Jacobian
> >     const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, J_00 / detJ}};
> >
> >     // Declare transformation matrix
> >     // Declare pointer to two dimensional array and initialise
> >     double **transform = new double *[num_derivatives];
> >
> >     for (unsigned int j = 0; j < num_derivatives; j++)
> >     {
> >       transform[j] = new double [num_derivatives];
> >       for (unsigned int k = 0; k < num_derivatives; k++)
> >         transform[j][k] = 1;
> >     }
> >
> >     // Construct transformation matrix
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         for (unsigned int k = 0; k < n; k++)
> >           transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
> >       }
> >     }
> >
> >     // Reset values
> >     for (unsigned int j = 0; j < 1*num_derivatives; j++)
> >       values[j] = 0;
> >
> >     // Map degree of freedom to element degree of freedom
> >     const unsigned int dof = i;
> >
> >     // Generate scalings
> >     const double scalings_y_0 = 1;
> >     const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
> >
> >     // Compute psitilde_a
> >     const double psitilde_a_0 = 1;
> >     const double psitilde_a_1 = x;
> >
> >     // Compute psitilde_bs
> >     const double psitilde_bs_0_0 = 1;
> >     const double psitilde_bs_0_1 = 1.5*y + 0.5;
> >     const double psitilde_bs_1_0 = 1;
> >
> >     // Compute basisvalues
> >     const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
> >     const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
> >     const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
> >
> >     // Table(s) of coefficients
> >     static const double coefficients0[3][3] = \
> >     {{0.471404520791032, -0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0.288675134594813, -0.166666666666667},
> >     {0.471404520791032, 0, 0.333333333333333}};
> >
> >     // Interesting (new) part
> >     // Tables of derivatives of the polynomial base (transpose)
> >     static const double dmats0[3][3] = \
> >     {{0, 0, 0},
> >     {4.89897948556636, 0, 0},
> >     {0, 0, 0}};
> >
> >     static const double dmats1[3][3] = \
> >     {{0, 0, 0},
> >     {2.44948974278318, 0, 0},
> >     {4.24264068711928, 0, 0}};
> >
> >     // Compute reference derivatives
> >     // Declare pointer to array of derivatives on FIAT element
> >     double *derivatives = new double [num_derivatives];
> >
> >     // Declare coefficients
> >     double coeff0_0 = 0;
> >     double coeff0_1 = 0;
> >     double coeff0_2 = 0;
> >
> >     // Declare new coefficients
> >     double new_coeff0_0 = 0;
> >     double new_coeff0_1 = 0;
> >     double new_coeff0_2 = 0;
> >
> >     // Loop possible derivatives
> >     for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
> >     {
> >       // Get values from coefficients array
> >       new_coeff0_0 = coefficients0[dof][0];
> >       new_coeff0_1 = coefficients0[dof][1];
> >       new_coeff0_2 = coefficients0[dof][2];
> >
> >       // Loop derivative order
> >       for (unsigned int j = 0; j < n; j++)
> >       {
> >         // Update old coefficients
> >         coeff0_0 = new_coeff0_0;
> >         coeff0_1 = new_coeff0_1;
> >         coeff0_2 = new_coeff0_2;
> >
> >         if(combinations[deriv_num][j] == 0)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
> >           new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
> >           new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
> >         }
> >         if(combinations[deriv_num][j] == 1)
> >         {
> >           new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
> >           new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
> >           new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
> >         }
> >
> >       }
> >       // Compute derivatives on reference element as dot product of coefficients and basisvalues
> >       derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
> >     }
> >
> >     // Transform derivatives back to physical element
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       for (unsigned int col = 0; col < num_derivatives; col++)
> >       {
> >         values[row] += transform[row][col]*derivatives[col];
> >       }
> >     }
> >     // Delete pointer to array of derivatives on FIAT element
> >     delete [] derivatives;
> >
> >     // Delete pointer to array of combinations of derivatives and transform
> >     for (unsigned int row = 0; row < num_derivatives; row++)
> >     {
> >       delete [] combinations[row];
> >       delete [] transform[row];
> >     }
> >
> >     delete [] combinations;
> >     delete [] transform;
> >   }
> >
> >   /// Evaluate order n derivatives of all basis functions at given point in cell
> >   virtual void evaluate_basis_derivatives_all(unsigned int n,
> >                                               double* values,
> >                                               const double* coordinates,
> >                                               const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("The vectorised version of evaluate_basis_derivatives() is not yet implemented.");
> >   }
> >
> >   /// Evaluate linear functional for dof i on the function f
> >   virtual double evaluate_dof(unsigned int i,
> >                               const ufc::function& f,
> >                               const ufc::cell& c) const
> >   {
> >     // The reference points, direction and weights:
> >     static const double X[3][1][2] = {{{0, 0}}, {{1, 0}}, {{0, 1}}};
> >     static const double W[3][1] = {{1}, {1}, {1}};
> >     static const double D[3][1][1] = {{{1}}, {{1}}, {{1}}};
> >
> >     const double * const * x = c.coordinates;
> >     double result = 0.0;
> >     // Iterate over the points:
> >     // Evaluate basis functions for affine mapping
> >     const double w0 = 1.0 - X[i][0][0] - X[i][0][1];
> >     const double w1 = X[i][0][0];
> >     const double w2 = X[i][0][1];
> >
> >     // Compute affine mapping y = F(X)
> >     double y[2];
> >     y[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
> >     y[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
> >
> >     // Evaluate function at physical points
> >     double values[1];
> >     f.evaluate(values, y, c);
> >
> >     // Map function values using appropriate mapping
> >     // Affine map: Do nothing
> >
> >     // Note that we do not map the weights (yet).
> >
> >     // Take directional components
> >     for(int k = 0; k < 1; k++)
> >       result += values[k]*D[i][0][k];
> >     // Multiply by weights
> >     result *= W[i][0];
> >
> >     return result;
> >   }
> >
> >   /// Evaluate linear functionals for all dofs on the function f
> >   virtual void evaluate_dofs(double* values,
> >                              const ufc::function& f,
> >                              const ufc::cell& c) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Interpolate vertex values from dof values
> >   virtual void interpolate_vertex_values(double* vertex_values,
> >                                          const double* dof_values,
> >                                          const ufc::cell& c) const
> >   {
> >     // Evaluate at vertices and use affine mapping
> >     vertex_values[0] = dof_values[0];
> >     vertex_values[1] = dof_values[1];
> >     vertex_values[2] = dof_values[2];
> >   }
> >
> >   /// Return the number of sub elements (for a mixed element)
> >   virtual unsigned int num_sub_elements() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new finite element for sub element i (for a mixed element)
> >   virtual ufc::finite_element* create_sub_element(unsigned int i) const
> >   {
> >     return new poisson_auxiliary_1_finite_element_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for a local-to-global mapping of
> > /// degrees of freedom (dofs).
> >
> >
> > class poisson_auxiliary_1_dof_map_0: public ufc::dof_map
> > {
> > private:
> >
> >   unsigned int __global_dimension;
> >
> >
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_1_dof_map_0() :ufc::dof_map()
> >   {
> >     __global_dimension = 0;
> >   }
> >   /// Destructor
> >   virtual ~poisson_auxiliary_1_dof_map_0()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the dof map
> >   virtual const char* signature() const
> >   {
> >     return "FFC dof map for FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1)";
> >   }
> >
> >   /// Return true iff mesh entities of topological dimension d are needed
> >   virtual bool needs_mesh_entities(unsigned int d) const
> >   {
> >     switch ( d )
> >     {
> >     case 0:
> >       return true;
> >       break;
> >     case 1:
> >       return false;
> >       break;
> >     case 2:
> >       return false;
> >       break;
> >     }
> >     return false;
> >   }
> >
> >   /// Initialize dof map for mesh (return true iff init_cell() is needed)
> >   virtual bool init_mesh(const ufc::mesh& m)
> >   {
> >     __global_dimension = m.num_entities[0];
> >     return false;
> >   }
> >
> >   /// Initialize dof map for given cell
> >   virtual void init_cell(const ufc::mesh& m,
> >                          const ufc::cell& c)
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Finish initialization of dof map for cells
> >   virtual void init_cell_finalize()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return the dimension of the global finite element function space
> >   virtual unsigned int global_dimension() const
> >   {
> >     return __global_dimension ;
> >   }
> >
> >   /// Return the dimension of the local finite element function space for a cell
> >   virtual unsigned int local_dimension(const ufc::cell& c) const
> >   {
> >     return 3;
> >   }
> >
> >   /// Return the maximum dimension of the local finite element function space
> >   virtual unsigned int max_local_dimension() const
> >   {
> >     return 3;
> >   }
> >
> >
> >   // Return the geometric dimension of the coordinates this dof map provides
> >   virtual unsigned int geometric_dimension() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs on each cell facet
> >   virtual unsigned int num_facet_dofs() const
> >   {
> >     return 2;
> >   }
> >
> >   /// Return the number of dofs associated with each cell entity of dimension d
> >   virtual unsigned int num_entity_dofs(unsigned int d) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the local-to-global mapping of dofs on a cell
> >   virtual void tabulate_dofs(unsigned int* dofs,
> >                              const ufc::mesh& m,
> >                              const ufc::cell& c) const
> >   {
> >     dofs[0] = c.entity_indices[0][0];
> >     dofs[1] = c.entity_indices[0][1];
> >     dofs[2] = c.entity_indices[0][2];
> >   }
> >
> >   /// Tabulate the local-to-local mapping from facet dofs to cell dofs
> >   virtual void tabulate_facet_dofs(unsigned int* dofs,
> >                                    unsigned int facet) const
> >   {
> >     switch ( facet )
> >     {
> >     case 0:
> >       dofs[0] = 1;
> >       dofs[1] = 2;
> >       break;
> >     case 1:
> >       dofs[0] = 0;
> >       dofs[1] = 2;
> >       break;
> >     case 2:
> >       dofs[0] = 0;
> >       dofs[1] = 1;
> >       break;
> >     }
> >   }
> >
> >   /// Tabulate the local-to-local mapping of dofs on entity (d, i)
> >   virtual void tabulate_entity_dofs(unsigned int* dofs,
> >                                     unsigned int d, unsigned int i) const
> >   {
> >     throw std::runtime_error("Not implemented (introduced in UFC v1.1).");
> >   }
> >
> >   /// Tabulate the coordinates of all dofs on a cell
> >   virtual void tabulate_coordinates(double** coordinates,
> >                                     const ufc::cell& c) const
> >   {
> >     const double * const * x = c.coordinates;
> >     coordinates[0][0] = x[0][0];
> >     coordinates[0][1] = x[0][1];
> >     coordinates[1][0] = x[1][0];
> >     coordinates[1][1] = x[1][1];
> >     coordinates[2][0] = x[2][0];
> >     coordinates[2][1] = x[2][1];
> >   }
> >
> >   /// Return the number of sub dof maps (for a mixed element)
> >   virtual unsigned int num_sub_dof_maps() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Create a new dof_map for sub dof map i (for a mixed element)
> >   virtual ufc::dof_map* create_sub_dof_map(unsigned int i) const
> >   {
> >     return new poisson_auxiliary_1_dof_map_0();
> >   }
> >
> > };
> >
> > /// This class defines the interface for the assembly of the global
> > /// tensor corresponding to a form with r + n arguments, that is, a
> > /// mapping
> > ///
> > ///     a : V1 x V2 x ... Vr x W1 x W2 x ... x Wn -> R
> > ///
> > /// with arguments v1, v2, ..., vr, w1, w2, ..., wn. The rank r
> > /// global tensor A is defined by
> > ///
> > ///     A = a(V1, V2, ..., Vr, w1, w2, ..., wn),
> > ///
> > /// where each argument Vj represents the application to the
> > /// sequence of basis functions of Vj and w1, w2, ..., wn are given
> > /// fixed functions (coefficients).
> >
> > class poisson_auxiliary_form_1: public ufc::form
> > {
> > public:
> >
> >   /// Constructor
> >   poisson_auxiliary_form_1() : ufc::form()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Destructor
> >   virtual ~poisson_auxiliary_form_1()
> >   {
> >     // Do nothing
> >   }
> >
> >   /// Return a string identifying the form
> >   virtual const char* signature() const
> >   {
> >     return "Auxiliary ufc::form to initialize standard functions, apply boundary conditions and obtain ufc::dof_map objects for continuous space(required for PUM objects) for a form containing discontinuous spaces.";
> >   }
> >
> >   /// Return the rank of the global tensor (r)
> >   virtual unsigned int rank() const
> >   {
> >     return 1;
> >   }
> >
> >   /// Return the number of coefficients (n)
> >   virtual unsigned int num_coefficients() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of cell integrals
> >   virtual unsigned int num_cell_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of exterior facet integrals
> >   virtual unsigned int num_exterior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Return the number of interior facet integrals
> >   virtual unsigned int num_interior_facet_integrals() const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new finite element for argument function i
> >   virtual ufc::finite_element* create_finite_element(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_auxiliary_1_finite_element_0();
> >       break;
> >     case 1:
> >       return new poisson_auxiliary_1_finite_element_0();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new dof map for argument function i
> >   virtual ufc::dof_map* create_dof_map(unsigned int i) const
> >   {
> >     switch ( i )
> >     {
> >     case 0:
> >       return new poisson_auxiliary_1_dof_map_0();
> >       break;
> >     case 1:
> >       return new poisson_auxiliary_1_dof_map_0();
> >       break;
> >     }
> >     return 0;
> >   }
> >
> >   /// Create a new cell integral on sub domain i
> >   virtual ufc::cell_integral* create_cell_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new exterior facet integral on sub domain i
> >   virtual ufc::exterior_facet_integral* create_exterior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> >   /// Create a new interior facet integral on sub domain i
> >   virtual ufc::interior_facet_integral* create_interior_facet_integral(unsigned int i) const
> >   {
> >     return 0;
> >   }
> >
> > };
> >
> > /// This class defines the interface for post-processing on vector x
> > /// to obtain x0, u and j where,
> > ///
> > /// - x is the solution vector containing standard and enriched degrees of freedom
> > /// defined on continuous/discontinuous space
> > /// - u is the standard part of solution vector defined on continuous space
> > /// - j is the enriched part pf solution vector defined on continuous space
> > /// - x0 is the result vector, equall to u + j, defined on continuous space
> > /// by considering enrichement function
> > //
> >
> > // Dolfin includes
> > #include <dolfin/common/NoDeleter.h>
> > #include <dolfin/mesh/Mesh.h>
> > #include <dolfin/fem/DofMap.h>
> > #include <dolfin/la/GenericVector.h>
> >
> > // PartitionOfUnity includes
> > #include <pum/PostProcess.h>
> > #include <pum/FunctionSpace.h>
> >
> > namespace     Poisson
> > {
> >   class PostProcess:  public pum::PostProcess
> >   {
> >     dolfin::Mesh& mesh;
> >     std::vector<const pum::GenericPUM*>& pums;
> >
> >   public:
> >
> >     /// Constructor
> >     PostProcess(dolfin::Mesh& mesh, std::vector<const pum::GenericPUM*>& pums): pum::PostProcess(mesh), mesh(mesh), pums(pums)
> >     {
> >       // Do nothing
> >     }
> >
> >     /// Destructor
> >     ~PostProcess()
> >     {
> >       // Do nothing
> >     }
> >
> >     /// Return a string identifying the underling element
> >     const char* signature() const
> >     {
> >       return "Interpolating results to the continuous space of MixedElement(*[FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), ElementRestriction(FiniteElement('Lagrange', Cell('triangle', 1, Space(2)), 1), Measure('surface', 0, None))], **{'value_shape': (2,) })";
> >     }
> >
> >     /// Obtain result vector 'x0' from solution vector 'x'
> >     void interpolate(const dolfin::GenericVector& x, dolfin::GenericVector& x0) const
> >     {
> >
> >       //poisson_1_dof_map_0_0 ufc_dof_map_0;
> >       //dolfin::DofMap dof_map_0(ufc_dof_map_0, mesh);
> >       //dolfin::DofMap dof_map_0(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0_0()), dolfin::reference_to_no_delete_pointer(mesh));
> >       dolfin::DofMap dof_map_0(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0_0()), mesh);
> >       unsigned int num_standard_dofs_0 = dof_map_0.global_dimension();
> >
> >       double value, h;
> >
> >       /// selecting standard degrees of freedom related to field 0 from the solution vector
> >
> >       double* values_0 = new double[num_standard_dofs_0];
> >       unsigned int* positions_0 = new unsigned int [num_standard_dofs_0];
> >
> >       for (unsigned int i = 0; i < num_standard_dofs_0; ++i)
> >         positions_0[i] = i;
> >
> >       x.get(values_0, num_standard_dofs_0, positions_0);
> >       x0.set(values_0, num_standard_dofs_0, positions_0);
> >
> >
> >
> >       /// selecting enriched degrees of freedom related to field 0 from the solution vector
> >       std::vector <std::vector<unsigned int> > enhanced_dof_maps_0;
> >       enhanced_dof_maps_0.resize(num_standard_dofs_0);
> >
> >       std::vector<unsigned int> enhanced_dof_values_0;
> >       enhanced_dof_values_0.resize(num_standard_dofs_0);
> >
> >       compute_enhanced_dof_maps(*pums[0], dof_map_0, enhanced_dof_maps_0);
> >       compute_enhanced_dof_values(*pums[0], dof_map_0, enhanced_dof_values_0);
> >
> >
> >       for (unsigned int i = 0; i != num_standard_dofs_0; ++i)
> >       {
> >         unsigned int pos = i;
> >
> >         for (std::vector<unsigned int>::const_iterator it = enhanced_dof_maps_0[i].begin();
> >                            it != enhanced_dof_maps_0[i].end(); ++it)
> >         {
> >           //h = pums[0]->enhanced_node_value(*it);
> >           h = enhanced_dof_values_0[i];
> >           unsigned int pos_n = *it + num_standard_dofs_0;
> >
> >           x.get(&value, 1, &pos_n);
> >           value *= h;
> >           x0.add(&value, 1, &pos);
> >         }
> >       }
> >
> >
> >     // memory clean up
> >
> >       delete[] values_0;
> >       delete[] positions_0;
> >
> >       x0.apply();
> >     }
> >
> >     /// Obtain continuous u and discontinuous j parts of solution vector 'x'
> >     void interpolate(const dolfin::GenericVector& x, dolfin::GenericVector& u, dolfin::GenericVector& j) const
> >     {
> >
> >       //poisson_1_dof_map_0_0 ufc_dof_map_0;
> >       //dolfin::DofMap dof_map_0(ufc_dof_map_0, mesh);
> >       //dolfin::DofMap dof_map_0(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0_0()), dolfin::reference_to_no_delete_pointer(mesh));
> >       dolfin::DofMap dof_map_0(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0_0()), mesh);
> >       unsigned int num_standard_dofs_0 = dof_map_0.global_dimension();
> >
> >       double value;
> >
> >       /// selecting standard degrees of freedom related to field 0 from the solution vector
> >       double* values_0 = new double[num_standard_dofs_0];
> >       unsigned int* positions_0 = new unsigned int [num_standard_dofs_0];
> >
> >       for (unsigned int i = 0; i < num_standard_dofs_0; ++i)
> >         positions_0[i] = i;
> >
> >       x.get(values_0, num_standard_dofs_0, positions_0);
> >       u.set(values_0, num_standard_dofs_0, positions_0);
> >
> >
> >
> >       /// selecting enriched degrees of freedom related to field 0 from the solution vector
> >       std::vector <std::vector<unsigned int> > enhanced_dof_maps_0;
> >       enhanced_dof_maps_0.resize(num_standard_dofs_0);
> >
> >       compute_enhanced_dof_maps(*pums[0], dof_map_0, enhanced_dof_maps_0);
> >
> >       for (unsigned int i = 0; i != num_standard_dofs_0; ++i)
> >       {
> >         unsigned int pos = i ;
> >
> >         for (std::vector<unsigned int>::const_iterator it = enhanced_dof_maps_0[i].begin();
> >                            it != enhanced_dof_maps_0[i].end(); ++it)
> >         {
> >           unsigned int pos_n = *it + num_standard_dofs_0;
> >
> >           x.get(&value, 1, &pos_n);
> >           j.set(&value, 1, &pos);
> >        }
> >       }
> >
> >
> >     // memory clean up
> >
> >       delete[] values_0;
> >       delete[] positions_0;
> >
> >       u.apply();
> >       j.apply();
> >     }
> >
> >   };
> > }
> >
> > // DOLFIN wrappers
> >
> > // Standard library includes
> > #include <string>
> >
> > // DOLFIN includes
> > #include <dolfin/common/NoDeleter.h>
> > #include <dolfin/fem/FiniteElement.h>
> > #include <dolfin/fem/DofMap.h>
> > #include <dolfin/fem/Form.h>
> > #include <dolfin/function/FunctionSpace.h>
> > #include <dolfin/function/Function.h>
> > #include <dolfin/function/GenericFunction.h>
> > #include <dolfin/function/CoefficientAssigner.h>
> >
> > namespace Poisson
> > {
> >
> > class CoefficientSpace_f: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   CoefficientSpace_f(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_1_dof_map_1()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_f(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_1()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_1()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_f(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_1()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   CoefficientSpace_f(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_1()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~CoefficientSpace_f()
> >   {
> >   }
> >
> > };
> >
> > class CoefficientSpace_g: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   CoefficientSpace_g(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_1_dof_map_2()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_g(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_2()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_2()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_g(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_2()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   CoefficientSpace_g(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_2()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~CoefficientSpace_g()
> >   {
> >   }
> >
> > };
> >
> > class CoefficientSpace_k: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   CoefficientSpace_k(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_0_dof_map_2()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_k(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_2()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_2()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_k(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_2()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   CoefficientSpace_k(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_2()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_2()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~CoefficientSpace_k()
> >   {
> >   }
> >
> > };
> >
> > class CoefficientSpace_w: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   CoefficientSpace_w(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_3()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_0_dof_map_3()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_w(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_3()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_3()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   CoefficientSpace_w(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_3()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_3()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   CoefficientSpace_w(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_3()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_3()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~CoefficientSpace_w()
> >   {
> >   }
> >
> > };
> >
> > class Form_0_FunctionSpace_0: public pum::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_0_FunctionSpace_0(const dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_0_dof_map_0(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_0(dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >     pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_0()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_0(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_0(boost::shared_ptr<dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_0(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_0(boost::shared_ptr<const dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_0(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_0_FunctionSpace_0()
> >   {
> >   }
> >
> > };
> >
> > class Form_0_FunctionSpace_1: public pum::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_0_FunctionSpace_1(const dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_0_dof_map_1(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_1(dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >     pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_1()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_1(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_1(boost::shared_ptr<dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_1(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_0_FunctionSpace_1(boost::shared_ptr<const dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_0_finite_element_1()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_0_dof_map_1(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_0_FunctionSpace_1()
> >   {
> >   }
> >
> > };
> >
> > typedef CoefficientSpace_k Form_0_FunctionSpace_2;
> >
> > typedef CoefficientSpace_w Form_0_FunctionSpace_3;
> >
> > class Form_0: public dolfin::Form
> > {
> > public:
> >
> >   // Constructor
> >   Form_0(const pum::FunctionSpace& V0, const pum::FunctionSpace& V1):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >     _function_spaces[1] = reference_to_no_delete_pointer(V1);
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_0(const pum::FunctionSpace& V0, const pum::FunctionSpace& V1, dolfin::GenericFunction & k, dolfin::GenericFunction & w):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >     _function_spaces[1] = reference_to_no_delete_pointer(V1);
> >
> >     this->k = k;
> >     this->w = w;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_0(const pum::FunctionSpace& V0, const pum::FunctionSpace& V1, boost::shared_ptr<dolfin::GenericFunction> k, boost::shared_ptr<dolfin::GenericFunction> w):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >     _function_spaces[1] = reference_to_no_delete_pointer(V1);
> >
> >     this->k = *k;
> >     this->w = *w;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_0(boost::shared_ptr<const pum::FunctionSpace> V0, boost::shared_ptr<const pum::FunctionSpace> V1):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >     _function_spaces[1] = V1;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0->pums));
> >   }
> >
> >   // Constructor
> >   Form_0(boost::shared_ptr<const pum::FunctionSpace> V0, boost::shared_ptr<const pum::FunctionSpace> V1, dolfin::GenericFunction & k, dolfin::GenericFunction & w):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >     _function_spaces[1] = V1;
> >
> >     this->k = k;
> >     this->w = w;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0->pums));
> >   }
> >
> >   // Constructor
> >   Form_0(boost::shared_ptr<const pum::FunctionSpace> V0, boost::shared_ptr<const pum::FunctionSpace> V1, boost::shared_ptr<dolfin::GenericFunction> k, boost::shared_ptr<dolfin::GenericFunction> w):
> >     dolfin::Form(2, 2), k(*this, 0), w(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >     _function_spaces[1] = V1;
> >
> >     this->k = *k;
> >     this->w = *w;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_0(V0->pums));
> >   }
> >
> >   // Destructor
> >   ~Form_0()
> >   {}
> >
> >   /// Return the number of the coefficient with this name
> >   virtual dolfin::uint coefficient_number(const std::string& name) const
> >   {
> >     if(name == "k") return 0;
> >     else if(name == "w") return 1;
> >     dolfin::error("Invalid coefficient.");
> >     return 0;
> >   }
> >
> >   /// Return the name of the coefficient with this number
> >   virtual std::string coefficient_name(dolfin::uint i) const
> >   {
> >     switch(i)
> >     {
> >       case 0: return "k";
> >       case 1: return "w";
> >     }
> >     dolfin::error("Invalid coefficient.");
> >     return "unnamed";
> >   }
> >
> >   // Typedefs
> >   typedef Form_0_FunctionSpace_0 TestSpace;
> >   typedef Form_0_FunctionSpace_1 TrialSpace;
> >   typedef Form_0_FunctionSpace_2 CoefficientSpace_k;
> >   typedef Form_0_FunctionSpace_3 CoefficientSpace_w;
> >
> >   // Coefficients
> >   dolfin::CoefficientAssigner k;
> >   dolfin::CoefficientAssigner w;
> > };
> >
> > class Form_1_FunctionSpace_0: public pum::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_1_FunctionSpace_0(const dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_1_dof_map_0(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_1_FunctionSpace_0(dolfin::Mesh & mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >     pum::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_0()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0(pums)), mesh)) ,pums)
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_1_FunctionSpace_0(boost::shared_ptr<dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_1_FunctionSpace_0(boost::shared_ptr<const dolfin::Mesh> mesh ,const std::vector<const pum::GenericPUM*>& pums):
> >       pum::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_1_dof_map_0(pums)), *mesh)) ,pums)
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_1_FunctionSpace_0()
> >   {
> >   }
> >
> > };
> >
> > typedef CoefficientSpace_f Form_1_FunctionSpace_1;
> >
> > typedef CoefficientSpace_g Form_1_FunctionSpace_2;
> >
> > class Form_1: public dolfin::Form
> > {
> > public:
> >
> >   // Constructor
> >   Form_1(const pum::FunctionSpace& V0):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_1(const pum::FunctionSpace& V0, dolfin::GenericFunction & f, dolfin::GenericFunction & g):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >
> >     this->f = f;
> >     this->g = g;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_1(const pum::FunctionSpace& V0, boost::shared_ptr<dolfin::GenericFunction> f, boost::shared_ptr<dolfin::GenericFunction> g):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >
> >     this->f = *f;
> >     this->g = *g;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0.pums));
> >   }
> >
> >   // Constructor
> >   Form_1(boost::shared_ptr<const pum::FunctionSpace> V0):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0->pums));
> >   }
> >
> >   // Constructor
> >   Form_1(boost::shared_ptr<const pum::FunctionSpace> V0, dolfin::GenericFunction & f, dolfin::GenericFunction & g):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >
> >     this->f = f;
> >     this->g = g;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0->pums));
> >   }
> >
> >   // Constructor
> >   Form_1(boost::shared_ptr<const pum::FunctionSpace> V0, boost::shared_ptr<dolfin::GenericFunction> f, boost::shared_ptr<dolfin::GenericFunction> g):
> >     dolfin::Form(1, 2), f(*this, 0), g(*this, 1)
> >   {
> >     _function_spaces[0] = V0;
> >
> >     this->f = *f;
> >     this->g = *g;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_form_1(V0->pums));
> >   }
> >
> >   // Destructor
> >   ~Form_1()
> >   {}
> >
> >   /// Return the number of the coefficient with this name
> >   virtual dolfin::uint coefficient_number(const std::string& name) const
> >   {
> >     if(name == "f") return 0;
> >     else if(name == "g") return 1;
> >     dolfin::error("Invalid coefficient.");
> >     return 0;
> >   }
> >
> >   /// Return the name of the coefficient with this number
> >   virtual std::string coefficient_name(dolfin::uint i) const
> >   {
> >     switch(i)
> >     {
> >       case 0: return "f";
> >       case 1: return "g";
> >     }
> >     dolfin::error("Invalid coefficient.");
> >     return "unnamed";
> >   }
> >
> >   // Typedefs
> >   typedef Form_1_FunctionSpace_0 TestSpace;
> >   typedef Form_1_FunctionSpace_1 CoefficientSpace_f;
> >   typedef Form_1_FunctionSpace_2 CoefficientSpace_g;
> >
> >   // Coefficients
> >   dolfin::CoefficientAssigner f;
> >   dolfin::CoefficientAssigner g;
> > };
> >
> > // Class typedefs
> > typedef Form_0 BilinearForm;
> > typedef Form_1 LinearForm;
> > typedef Form_0::TestSpace FunctionSpace;
> >
> > } // namespace Poisson
> >
> > // DOLFIN wrappers
> >
> > // Standard library includes
> > #include <string>
> >
> > // DOLFIN includes
> > #include <dolfin/common/NoDeleter.h>
> > #include <dolfin/fem/FiniteElement.h>
> > #include <dolfin/fem/DofMap.h>
> > #include <dolfin/fem/Form.h>
> > #include <dolfin/function/FunctionSpace.h>
> > #include <dolfin/function/Function.h>
> > #include <dolfin/function/GenericFunction.h>
> > #include <dolfin/function/CoefficientAssigner.h>
> >
> > namespace Poisson
> > {
> >
> > class Form_auxiliary_0_FunctionSpace_auxiliary_0: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_0(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_auxiliary_0_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_0(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_0(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_0(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_auxiliary_0_FunctionSpace_auxiliary_0()
> >   {
> >   }
> >
> > };
> >
> > class Form_auxiliary_0_FunctionSpace_auxiliary_1: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_1(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_auxiliary_0_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_1(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_1(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_auxiliary_0_FunctionSpace_auxiliary_1(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_0_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_0_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_auxiliary_0_FunctionSpace_auxiliary_1()
> >   {
> >   }
> >
> > };
> >
> > class Form_auxiliary_0: public dolfin::Form
> > {
> > public:
> >
> >   // Constructor
> >   Form_auxiliary_0(const dolfin::FunctionSpace& V0, const dolfin::FunctionSpace& V1):
> >     dolfin::Form(2, 0)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >     _function_spaces[1] = reference_to_no_delete_pointer(V1);
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_auxiliary_form_0());
> >   }
> >
> >   // Constructor
> >   Form_auxiliary_0(boost::shared_ptr<const dolfin::FunctionSpace> V0, boost::shared_ptr<const dolfin::FunctionSpace> V1):
> >     dolfin::Form(2, 0)
> >   {
> >     _function_spaces[0] = V0;
> >     _function_spaces[1] = V1;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_auxiliary_form_0());
> >   }
> >
> >   // Destructor
> >   ~Form_auxiliary_0()
> >   {}
> >
> >   /// Return the number of the coefficient with this name
> >   virtual dolfin::uint coefficient_number(const std::string& name) const
> >   {
> >     dolfin::error("No coefficients.");
> >     return 0;
> >   }
> >
> >   /// Return the name of the coefficient with this number
> >   virtual std::string coefficient_name(dolfin::uint i) const
> >   {
> >     dolfin::error("No coefficients.");
> >     return "unnamed";
> >   }
> >
> >   // Typedefs
> >   typedef Form_auxiliary_0_FunctionSpace_auxiliary_0 TestSpace;
> >   typedef Form_auxiliary_0_FunctionSpace_auxiliary_1 TrialSpace;
> >
> >   // Coefficients
> > };
> >
> > class Form_auxiliary_1_FunctionSpace_auxiliary_0: public dolfin::FunctionSpace
> > {
> > public:
> >
> >
> >   Form_auxiliary_1_FunctionSpace_auxiliary_0(const dolfin::Mesh & mesh):
> >       dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement
> >                             (boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>
> >                             (new poisson_auxiliary_1_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_1_FunctionSpace_auxiliary_0(dolfin::Mesh & mesh):
> >     dolfin::FunctionSpace(dolfin::reference_to_no_delete_pointer(mesh),
> >                           boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_1_finite_element_0()))),
> >                           boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_1_dof_map_0()), mesh)))
> >   {
> >     // Do nothing
> >   }
> >
> >   Form_auxiliary_1_FunctionSpace_auxiliary_0(boost::shared_ptr<dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_1_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >   Form_auxiliary_1_FunctionSpace_auxiliary_0(boost::shared_ptr<const dolfin::Mesh> mesh):
> >       dolfin::FunctionSpace(mesh,
> >                             boost::shared_ptr<const dolfin::FiniteElement>(new dolfin::FiniteElement(boost::shared_ptr<ufc::finite_element>(new poisson_auxiliary_1_finite_element_0()))),
> >                             boost::shared_ptr<const dolfin::DofMap>(new dolfin::DofMap(boost::shared_ptr<ufc::dof_map>(new poisson_auxiliary_1_dof_map_0()), *mesh)))
> >   {
> >       // Do nothing
> >   }
> >
> >
> >   ~Form_auxiliary_1_FunctionSpace_auxiliary_0()
> >   {
> >   }
> >
> > };
> >
> > class Form_auxiliary_1: public dolfin::Form
> > {
> > public:
> >
> >   // Constructor
> >   Form_auxiliary_1(const dolfin::FunctionSpace& V0):
> >     dolfin::Form(1, 0)
> >   {
> >     _function_spaces[0] = reference_to_no_delete_pointer(V0);
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_auxiliary_form_0());
> >   }
> >
> >   // Constructor
> >   Form_auxiliary_1(boost::shared_ptr<const dolfin::FunctionSpace> V0):
> >     dolfin::Form(1, 0)
> >   {
> >     _function_spaces[0] = V0;
> >
> >     _ufc_form = boost::shared_ptr<const ufc::form>(new poisson_auxiliary_form_0());
> >   }
> >
> >   // Destructor
> >   ~Form_auxiliary_1()
> >   {}
> >
> >   /// Return the number of the coefficient with this name
> >   virtual dolfin::uint coefficient_number(const std::string& name) const
> >   {
> >     dolfin::error("No coefficients.");
> >     return 0;
> >   }
> >
> >   /// Return the name of the coefficient with this number
> >   virtual std::string coefficient_name(dolfin::uint i) const
> >   {
> >     dolfin::error("No coefficients.");
> >     return "unnamed";
> >   }
> >
> >   // Typedefs
> >   typedef Form_auxiliary_1_FunctionSpace_auxiliary_0 TestSpace;
> >
> >   // Coefficients
> > };
> >
> > // Class typedefs
> > typedef Form_auxiliary_0 BilinearForm_auxiliary;
> > typedef Form_auxiliary_1 LinearForm_auxiliary;
> > typedef Form_auxiliary_0::TestSpace FunctionSpace_auxiliary;
> >
> > } // namespace Poisson
> >
> > #endif
> 
> > # Copyright (C) 2008-2009 Mehdi Nikbakht and Garth N. Wells.
> > # Licensed under the GNU GPL Version 3.0 or any later version.
> > #
> > # The bilinear form a(v, u) and linear form L(v) for
> > # Poisson's equation with discontinuities.
> > #
> > # Compile this form with FFC: ffc-pum -l dolfin Poisson.ufl
> > #
> >
> > elem_cont = FiniteElement("CG", triangle, 1)
> > elem_discont = ElementRestriction(elem_cont, dc) # or ec[dc]
> >
> > element = elem_cont + elem_discont
> >
> > (vc, vd) = TestFunctions(element)
> > (uc, ud) = TrialFunctions(element)
> >
> > v = vc + vd
> > u = uc + ud
> >
> >
> > k  = Constant(triangle)
> > f  = Coefficient(elem_cont)
> > w  = Coefficient(elem_cont)
> > g  = Coefficient(elem_cont)
> >
> > a = w*dot(grad(v), grad(u))*dx + k*inner(vd, ud)*dc #+ (u*v)('-')*dS
> > L = v*f*dx - v*g*ds
> >
>