Algebraic Multilevel Preconditioning Based on Element Agglomeration

Johannes K. Kraus

University of Leoben
Franz Josef Strasse 18
A-8700 Leoben


We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of 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 matrices 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 offers the opportunity to construct a new kind of incomplete LU factorization of the pivot matrix at every level, which is done by means of a slightly modified assembling process. Based on these components an algebraic multilevel preconditioner can be defined for more general SPD matrices. The method can also be applied to systems of PDEs. A numerical analysis shows its efficiency and robustness.