Optimal v-cycle algebraic multilevel preconditioning

Abstract

We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete second-order elliptic PDEs, optimal convergence properties are proved for such basic two-level schemes under mild assumptions on the PDE coefficients, but their recursive use in a simple V-cycle algorithm does not generally lead to optimal order convergence. In the present paper, we analyse the combination of these techniques with a smoothing procedure much the same as the one used in standard multigrid algorithms, except that smoothing is not required on the finest grid. Theoretical results prove optimal convergence properties for the V-cycle under an assumption similar to the ‘approximation property’ of the classical multigrid convergence theory. On the other hand, numerical experiments made on both 2D and 3D problems show that the condition number is close to that of the two-level method. Further, the method appears robust in the presence of discontinuity and anisotropy, even when the material interfaces are not aligned with the coarse grid. Copyright © 1999 John Wiley & Sons, Ltd.