fenics team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Fri, Oct 22, 2010 at 03:12:47PM +0200, Marie Rognes wrote:
> On 30. sep. 2010 03:31, Douglas Arnold wrote:
> > FEniCS/UFL uses historical element names, like Raviart-Thomas,
> > Nedelec 1st kind H(div), etc.  These have to be learned, although
> > this is fortunately ameliorated somewhat by having some
> > synonyms, like N1curl.  It can also cause confusion sometimes,
> > as the names are not always used consistently in the literature
> > (and I think that "Discontinuous Lagrange" is a contradiction
> > in terms).
> >
> > However there is a systematic, consistent way to refer to
> > the majority of elements available, so I am making the
> > suggesting that this systematic form be added as
> > an additional option.  The systematic way is what is
> > used in the Finite Element Exterior Calculus, namely the
> > two families, which are in denoted in LaTeX by
> > $P_r\Lambda^k$ and $P_r^-\Lambda^k$, defined for
> > polynomial degree r=1,2,... and form degree k between
> > 0 and n on simplices of dimension n.  Thus I am suggesting
> > adding something like
> >
> >   V = FunctionSpace(mesh, "P Lambda", r, k)
> > and
> >   V = FunctionSpace(mesh, "P- Lambda", r, k)
> >
> > The translation to current names is:
> >
> >   {"P Lambda", r, 0} and {"P- Lambda", r, 0} coincide and
> >      are equal to {"Lagrange", r}
> >   {"P Lambda", r, n} is the same as {"Discontinuous Lagrange", r}
> >   {"P- Lambda", r, n} is the same as {"Discontinuous Lagrange", r-1}
> >   {"P- Lambda", r, 1} is {"Nedelec 1st kind H(curl)", r}
> >   {"P Lambda", r, 1} is {"Nedelec 2nd kind H(curl)", r}
> >   {"P- Lambda", r, n-1} is {"Nedelec 1st kind H(div)", r}
> >   {"P Lambda", r, n-1} is {"Nedelec 2nd kind H(div)", r}
> >
> > I want to stress that there is nothing arbitrary about this.
> > The whole family of P Lambda and P- Lambda spaces can be
> > defined at once for all r, k, n.  This even works for n=1
> > and n>3 (although the latter are not yet implemented in
> > FEniCS).
> >
> > There is a small issue in 2D where n-1 = 1, so the last
> > two cases conflict with the preceding two.  This is the
> > choice between the rotated Raviart-Thomas or Brezzi-Douglas-Marini
> > elements or the usual ones in 2D.  I would suggest always
> > thinking of 1-forms as the images of gradients, and hence
> > 1-forms always correspond to H(curl), also in 2D.
>
>
> This has now been implemented in UFL using the notation
>
>    V = FiniteElement("P Lambda", cell, r, k)
>    V = FiniteElement("P- Lambda", cell, r, k)
>
> where cell can be one of [interval, triangle, tetrahedron], r >= 1 [cf
> footnote 1] and k in [0, n] where n is the topological dimension of the
> cell.
>
> While at it, I also added an "exterior_derivative" operator, mapping to
> grad, curl, div, id based on the finite element space. An example of how
> this can be used in DOLFIN is included below. (Hopefully, the original
> requester will recognize the problem.)
>
> ------------------------------------------------------------------------------------------------------------------------
> from dolfin import *
> from ufl import exterior_derivative as d
>
> meshes = [UnitInterval(2), UnitSquare(2, 2), UnitCube(2, 2, 2)]
> families = ["P Lambda", "P- Lambda"]
>
> r = 1
> for (n, mesh) in enumerate(meshes):
>   for family in families:
>       for k in range(1, n+1):
>
>           S = FunctionSpace(mesh, "P- Lambda", r+1, k-1)
>           V = FunctionSpace(mesh, "P Lambda", r, k)
>           W = S * V
>
>           (s, u) = TrialFunctions(W)
>           (t, v) = TestFunctions(W)
>
>           a  = (- s, t) + (d(t), u) + (d(s), v) + (d(u), d(v))
>           A = assemble(a)
>           info(A)

Very nice! :-)

--
Anders