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"