fenics team mailing list archive

[Marie Rognes <meg@xxxxxxxxx>]

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)

------------------------------------------------------------------------------------------------------------------------

Footnotes:

[1] "P Lambda", "tetrahedron", r, 1 (aka Nedelec 2nd kind H(curl)
elements) is only supported by FIAT for r == 1.

--
Marie