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

Justin Chang jychang48 at gmail.com
Mon Jun 1 01:12:25 CDT 2015

There are a few papers that discuss this modified/augmented Taylor-Hood
elements for Stokes equations in detail (e.g.,
http://link.springer.com/article/10.1007%2Fs10915-011-9549-4). From what I
have seem, it seems people primarily use this to ensure local mass
conservation while attaining the desirable qualities of the TH element.
Lately I have seen this element used in many FEniCS and Deal.II
applications (and it's also very easy to implement, just a few additional
lines of code), which is why I had wanted to experiment with this myself
(if possible) using DMPlex.

On Sun, May 31, 2015 at 7:59 PM, Jed Brown <jed at jedbrown.org> wrote:

> Justin Chang <jychang48 at gmail.com> writes:
> > I am referring to P2 / (P1 + P0) elements, I think this is the correct
> way
> > of expressing it. Some call it modified Taylor Hood, others call it
> > something else, but it's not Crouzeix-Raviart elements.
> Okay, thanks.  This pressure space is not a disjoint union (the constant
> exists in both spaces) and thus the obvious "basis" is actually not
> linearly independent.  I presume that people using this element do some
> "pinning" (like set one cell "average" to zero) instead of enforcing a
> unique expression via a Lagrange multiplier (which would involve a dense
> row and column).  That may contribute to ill conditioning and in any
> case, would make domain decomposition or multigrid preconditioners more
> technical.  Do you know of anything explaining why the method is not
> very widely used (e.g., in popular software packages, finite element
> books, etc.)?
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