# [petsc-users] Modified Taylor-Hood elements with piece-wise constant pressure for Stokes equation

Matthew Knepley knepley at gmail.com
Tue Jun 2 07:37:35 CDT 2015

```On Tue, Jun 2, 2015 at 7:18 AM, Lawrence Mitchell <
lawrence.mitchell at imperial.ac.uk> wrote:

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> On 02/06/15 13:14, Matthew Knepley wrote:
> > On Tue, Jun 2, 2015 at 6:13 AM, Justin Chang <jychang48 at gmail.com
> > <mailto:jychang48 at gmail.com>> wrote:
> >
> > In FEniCS's Stokes example (example 19), one defines the
> > Taylor-Hood function spaces with these three lines:
> >
> > V = VectorFunctionSpace(mesh, "CG", 2) Q = FunctionSpace(mesh,
> > "CG", 1) W = V * Q
> >
> > To implement P2/(P1+P0), all we gotta do is this:
> >
> > V = VectorFunctionSpace(mesh, "CG", 2) Q = FunctionSpace(mesh,
> > "CG", 1) P = FunctionSpace(mesh, "DG", 0) W = V * (Q + P)
> >
> >
> > So here you would need 4 dual basis vectors, which I am assuming
> > are:
> >
> > ev_(-1, -1), ev_(1, -1), ev_(-1, 1), ev_(-0.5, -0.5)
> >
> > where ev_(x, y) is the point evaluation functional at (x, y). Then
> > you need some basis for the primal space, which naively is
> >
> > 1, x, y, 1
> >
> > As you can see, this basis in linearly dependent, so the
> > Vandermonde matrix that FIAT constructs will be singular. The
> > construction of a nodal basis will fail.
> >
> > So Jed's question is, what are they actually doing internally?
>
> So-called "enriched" elements in FEniCS are not created with a nodal
> basis, instead te "basis" for the space Q + P is just the
> concatenation of the bases for Q and P separately and so tabulation of
> basis functions at points is just the concatentation of the tabulation
> of Q and that of P.
>

This construction appears to throw away unisolvence. In the Boffi paper,
they
use QR to solve the pressure Laplacian. This would put a damper on me
using the method.

Justin, how do they solve the system in the FEniCS example?

Thanks,

Matt

> Lawrence
>
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