# Multilevel Preconditioning Based on Element Agglomeration

## Priv.-Doz. Dr. Johannes Kraus

**Nov. 6, 2002, 4 p.m. HS 10**

We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of scalar elliptic PDEs. The method is based on element agglomeration, and, in particular, designed for Non-M matrices.

Granted that the element matrices at the fine-grid level are given, we further assume that we have access to some algorithm that performs a reasonable agglomeration of fine-grid elements at any given level. The coarse-grid element matrices are simply Schur complements computed from the locally assembled fine-grid element matrices, i.e., agglomerate matrices. Hence, these can be assembled to a global approximate Schur complement. The elimination of fine-degrees of freedom in the agglomerate matrices is done without neglecting any fill-in. This provides us the opportunity to construct a new kind of incomplete LU factorization for the pivot matrix at every level, which is done within a slightly modified assembling process.

Based on these components a powerful algebraic multilevel preconditioner can be defined for more general SPD matrices. A numerical analysis shows the efficiency and robustness of the new method.