Algebraic Multigrid Methods based on Element Stiffness Matrices

Stefan Reitzinger

Johannes Kepler University Linz
Spezialforschungsbereich SFB F013
"Numerical and Symbolic Scientific Computing"
Freistaedter Strasze 313, first floor
A-4040 Linz, Austria
 

Abstract

In order to solve large, linear, discrete systems with a sparse matrix structure an efficient solution strategy is required. The AMG method (introduced by Brandt/McCormick/Ruge) has proven to be an efficient and robust solver if the system is an s.p.d. M-matrix arising from an FE-discretization.
Unfortunately the "region of robustness" for s.p.d. M-matrices and general s.p.d. matrices is very fuzzy. For this reason we present two methods to deal with that problem. Therefore we assume access to the element stiffness matrices.

First, we present a method to preserve the M-matrix property such that the best possible spectral equivalent M-matrix to the original one is obtained. The efficiency of this technique is shown for anisotropic rectangles with bilinear FE-functions. AMG, with the spectral equivalent M-matrix with respect to the original stiffness matrix, is then applied as a preconditioner to the CG method.
In this case we can show numerically the robustness of this technique. Unfortunately the method will not work for triangles or tetrahedra.

Second, we suggest a method which acts locally on element patches.
Therefore, we use the Schur-complement on such local patches. Then it is straightforward to obtain  overlapping coarse grid operators and locally defined prolongation operators. With this technique we also get a kind of patch structure on the coarser levels. Thus the the algorithm can be applied recursively. We think that this method behaves also very robust, but it is not longer depending on the M-matrix property.
Finally numerical results are shown.


Acknowledgment: This work has been supported by the Austrian Science Foundation - "Fonds zur Förderung der wissenschaftlichen Forschung (FWF)" - SFB F013 "Numerical and Symbolic Scientific Computing"