Combinatorial Preconditioners for Scalar Elliptic Finite-elements Problems

We present a new preconditioner for linear systems arising from finite-elements discretizations of scalar elliptic partial differential equations (pde’s). The solver splits the collection {Ke} of element matrices into a subset E(t) of matrices that are approximable by diagonally-dominant matrices and a subset of matrices that are not approximable. The approximable Ke’s are approximated by diagonally-dominant matrices Le’s that are scaled and assembled to form a global diagonally-dominant matrix L = ∑ e∈E(t) αeLe. A combinatorial graph algorithm approximates L by another diagonally-dominant matrixM that is easier to factor. The sparsification M is scaled and added to the inapproximable elements; the sum γM+ ∑ e ∈E(t) Ke is factored and used as a preconditioner. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Experimental results show that on problems in which some of theKe’s are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.