An Algebraic Multilevel Approach for Highly Indefinite Systems of Equations

We will discuss algebraic multigrid techniques that address solving large sparse symmetric indefinite systems. In particular we consider the situation when the system is highly indefinite. Such cases arise e.g. from the discretization of the Helmholtz equation for high wave numbers or the Anderson model of localization.

As basis we mainly focus on three major aspects:

1. symmetric maximum weight matchings to increase the block diagonal dominance of the system,
2. inverse--based pivoting to drive the coarsening process and finally
3. filtering techniques to handle frequencies near zero eigenvalues.

These techniques are used within a multilevel framework and we will illustrate the resulting multilevel methods for selected numerical examples.

Joint work R. Römer (U. Warwick) and M. Grote, O. Schenk (both U. Basel).