Optimal Order Preconditioning of Finite Difference Matrices

Abstract

A new multilevel preconditioner is proposed for the iterative solution of linear systems whose coefficient matrix is a symmetric M-matrix arising from the discretization of a second-order elliptic PDE. It is based on a recursive block incomplete factorization of the matrix partitioned in a two-by-two block form, in which the submatrix related to the fine grid nodes is approximated by a modified incomplete (MILU) factorization and the Schur complement computed from a diagonal approximation of the same submatrix. A general algebraic analysis proves optimal order convergence under mild assumptions related to the quality of the approximations of the Schur complement on the one hand and of the fine grid submatrix on the other hand. This analysis does not require a red-black ordering of the fine grid nodes, nor a transformation of the matrices in hierarchical form. Considering more specifically 5 point finite difference approximations of two-dimensional problems, we prove that the spectrum of the preconditioned system is contained in the interval [0.46, 3], independently of possible (large) jumps or anisotropy in the PDE coefficients, as long as the latter are piecewise constant on the coarsest mesh. Numerical results illustrate the efficiency and the robustness of the proposed method.