[firedrake] Concatenation of function spaces?

[Lawrence Mitchell]

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On 22/07/15 07:31, Andrew McRae wrote:
> 1)  Yes, this can be done in Firedrake.  However, we consider the
> local element to be enriched, not the function space:
> 
> V1 = VectorFunctionSpace... Q_elt = FiniteElement("CG", interval,
> 1) P_elt = FiniteElement("DG", interval, 0) elt = Q_elt + P_elt V2
> = FunctionSpace(mesh, elt) W = V1 * V2
> 
> 2)  Is P1disc the same as P1DG?  "All polynomials of degree 1 with
> no continuity between cells"?  If so, none of us have implemented
> the P1disc elements on quadrilaterals yet in FIAT.  We have full
> support for the third column of http://femtable.org/, which has all
> the "tensor product construction" elements, but not the fourth.  If
> you want to try implementing P1disc yourself, we can hopefully
> provide support.

Indeed yes.

Note that P1disc provides some problems for the standard approach of
pulling back to a reference element and integrating.  Unless your mesh
is asymptotically affine you only get 1st order convergence in the L_2
norm of P.  This is because the mapped global space doesn't contain P_k.

See:

Matthies and Tobiska. The Inf-Sup Condition for the Mapped
$Q_k-P_{k-1}^{\mathrm{disc}}$ Element in Arbitrary Space Dimensions (2002)

and

Boffi and Gastaldi. On the quadrilateral $Q_2–P_1$ element for the
Stokes problem (2002)

For details.

Lawrence

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