Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

Abstract

Abstract. We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection {Ke} 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 Ke's are approximated by diagonally dominant matrices Le's that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning paradigms. Experimental results show that on problems in which some of the Ke'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.

Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners

“…In a third development, Avron et al [2] have shown how to get an approximation to element stiffness matrices that is optimal up to a constant factor. In other words, for an element stiffness matrix K t they find a diagonally dominant matrixK t such κ(K t ,K t ) ≤ c 1 κ(K t , K ′ t ), where K ′ t is any other symmetric diagonally dominant matrix of the correct size and c 1 depends only on p, d. Thus, their preconditioner could lead to a faster algorithm than ours since ours is not optimal in this sense.…”