Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

Based on a particular node ordering and corresponding block decomposition of the matrix we analyse an e cient, algebraic multilevel preconditioner for the iterative solution of nite element discretizations of elliptic boundary value problems. Thereby an analysis of a new version of block-factorization preconditioning methods is presented. The approximate factorization requires an approximation of the arising Schur complement matrix. In this paper we consider such approximations derived by the assembly of the local macro-element Schur complements. The method can be applied also for non-selfadjoint problems but for the derivation of condition number bounds we assume that the corresponding di erential operator is selfadjoint and positive de nite.