Preconditioning by approximate Schur complements on hierarchical grids

C. Brand

Abstract

We present an algebraic multilevel preconditioning strategy for difference matrices arising from discretization of three-dimensional boundary-value problems. Here we consider only V-cycles although an extension to other multigrid cycles may be useful. Based on a special ordering of the gridpoints the construction of our preconditioner is a (modified) incomplete factorization prozess. The Schur complements which occur during a blockwise elimination of the unknowns refer to a sequence of hierarchical grids and we approximate the corresponding operators by simpler ones until we reach a final coarsest grid. For the remaining unknowns the elimination is done exactly. This procedure results in a preconditioner which is well suited to parallel architectures.

For the 3-D Poisson modell problem with Dirichlet boundary conditions we show that the condition number of the preconditioned system is bounded by O(h^{− \frac{2}{3}} ) which is better than for the standard MILU-factorization with relative condition number O(h^{−1}).