Berkeley Lab Scientific Computing Seminar

The talk will present a new preconditioner for linear systems arising from finite-elements discretizations of scalar elliptic PDE's. The algebraic equations that we solve are Kx=b, where K=∑_eK_e is a sum of element matrices K_e. The solver splits the collection {K_e}  of element matrices into a subset of matrices that are approximable by diagonally-dominant matrices and a subset of matrices that are not approximable. The approximable K_e's are approximated by diagonally-dominant matrices L_e that are scaled and assembled to form a global diagonally-dominant matrix L. A combinatorial graph algorithm approximates L by another diagonally-dominant matrix M that is much easier to factor. The inapproximable element matrices are added to M and the sum is factored and used as a preconditioner. When all the element matrices are approximable (in particular, when they are all well-conditioned), which is often the case, the preconditioner is provably efficient. Experimental results show that on some problems, especially problems with some ill conditioned elements, the preconditioner is more effective than an algebraic multigrid solver and than an incomplete-factorization preconditioner.

(This new solver is different than the one I talked about at UCB in February.)

This is joint work with my students Haim Avron, Doron Chen, and Gil Shklarski.