Best preconditioner for mixed-poisson problem (RT0 elements)

Question

For a very large mixed-poisson problem with lowest order Raviart-Thomas elements (RT0), I plan on using an iterative solver. However, this kind of problem is not positive-definite (saddle point problem IIRC), thus standard preconditioners (like jacobi) will not converge well for large-problems.

I was told that the algebraic multigrid preconditioner is the one of the most common methods to achieving fast convergence for Krylov solvers. My question is, what would it look like for a mixed-poisson problem? Or if there are better preconditioning methods, do point them out to me.

Thanks!

Answer 1

Algebraic Multigrid (AMG) methods have the fundamental limitation that, by and large, their current implementations only work reliable and well for elliptic or similar problems. For saddle point problems, you need to reduce things to blocks for which the AMG methods work. This could include the top left (mass matrix) block of the mixed formulation. In the end, for block systems such as the one you have, the best preconditioners work by operating on individual blocks of your matrix.

I've collected some thoughts on this in lecture 38 at http://www.math.tamu.edu/~bangerth/videos.html . You can also look at the implementation of a (non-optimal) approach in lecture 21.