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

We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagonally dominant problem. Significant theory has already been developed for preconditioners in the diagonally dominant case. We show that the degradation in the quality of the preconditioner using our technique is only a small constant factor.