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

[Jed Brown]

Justin Chang <jychang48 at gmail.com> writes:
> I am not quite sure what you're asking for. Are you asking for how people
> actually implement this augmented TH? In other words, how the shape/basis
> functions for this mixed function space would look? 

Ciarlet's finite element definition requires a local approximation space
("P1+P0" in your case -- though P0 is a subset of P1 so this is actually
just P1) and a set of "nodes" (a dual basis, which will define the basis
functions used for computation and the continuity between elements).
The dual basis needs to be unisolvent, such that the Vandermonde matrix
is invertible.

Now what might that look like for "P1+P0"?

And if you bypass Ciarlet's construction and just try to pick some
"basis" functions, say {(1-x)/2, (1+x)/2, 1} for the 1D simplex [-1,1],
then you can express the constant f(x)=1 as u=(1,1,0), v=(0,0,1), or t*u
+ (1-t)*v for any real value t.  Clearly it fails to be a basis because
it's linearly dependent.

Can you still compute with it?  The answer is probably yes if you pin
some cell displacement, but that causes a bunch of somewhat unattractive
side-effects that I asked about in my earlier message.

Now you said it was "just a few additional lines of code", so maybe you
can explain how people are implementing it.

> I have only seen in some key note lectures and presentations at
> conferences briefly mentioning this P2/(P1+P0) element, as if it's the
> de facto discretization for Stokes flows. 

It's hardly "de facto".

> That said, even I am not too sure how this would look.

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