Algebraic Multigrid and Element Preconditioning

Stefan Reitzinger
University of Linz, Austria

Abstract

This paper presents a solution strategy for poor conditioned, large linear systems with a sparse matrix arising from an FE-discretization. Especially robustness has to be gained, if anisotropic elements are used. To achieve this, the AMG method of Ruge/Stuben is used. This algorithm is robust for M-matrices, but unfortunately the `region of robustness' between s.p.d. M-matrices and general s.p.d. matrices is very fuzzy . For this reason the so called element preconditioning technique is introduced to obtain a spectral equivalent M-matrix with respect to the original stiffness matrix. AMG, with the spectral equivalent M-matrix instead of the original stiffness matrix, is then applied as preconditioner to the conjugate gradient method.

Numerical studies are done for a magnetic shielding, a sandwich and a boundary layer problem in 2D(and 3D) which show the numerical robustness of the new preconditioning method.