From: Roy S. <roy...@ic...> - 2007-07-05 20:46:48
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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 |