Re: [Libmesh-users] finite volume

[Roy S. <roy...@ic...>]

On Thu, 5 Jul 2007, Tahar Amari wrote:

> I read your slides about divergence free finite element .
> It is very interesting.
> It sounds you did not need to use Raviart-Thomas finite element.

That's correct.

> Would you suggest me that I could be able to solve
> MHD equation, now a vector field B is divergence free
> without having to use H(div) or  H(rot) finite element,
> on Tetrahedra.

I don't think I would recommend it.  The trouble with using
divergence-free spaces generated from the curl operator is that in
three dimensions the curl operator's kernel is huge.  In 2D the kernel
is one dimensional and so it's easy to constrain away to produce a
positive definite system; in 3D you either need a very clever way of
constraining away the kernel or you need your solvers to be able to
handle highly underdefined systems of equations.  My graduate studies
committee claimed the problem was "too hard or impossible" in 3D,
which was an overstatement, but it was turning out to be difficult and
inefficient enough that I didn't argue about changing my dissertation
topic.

> Suppose I have a tetrahedral mesh, what kind of element could I use
> ?  Degree of freedom on the vertices of the tetrahedra ?
>
> Do you rewrite the equaitons with a potential vector
> B=rot(A) such that div B= 0 (your slide #3 writes u=rot(psi).

Yes; but unlike in 2D where psi is a scalar (with an implicit vector
direction of +z), in 3D A is a full vector.  Your finite element space
is then three scalar elements, $C^1$ elements if you don't want end up
with interface integrals in your weak formulation.  The only 3D C^1
elements in libMesh are the Hermite cubes, though, so if you needed to
use a tetrahedral mesh you would have to implement something like the
Alfeld-Awanou-Lai 4-split tetrahedral macroelement.

That last sentence was probably the only part of this email relevant
to libMesh as a whole, so if you've got more questions on the subject
we won't subject libmesh-users to the discussion.  If I haven't
discouraged you yet, send me a private email and I can dig through my
thesis bibliography for you; I think I've got a reference or two to
papers using similar techniques for MHD.
---
Roy