Macro-elementwise preconditioning methods

Extremely large scale problems, modelled by partial differential equations, arise in various applications, and must be solved by properly preconditioned iterative methods. Frequently, the corresponding medium is heterogeneous. Recursively constructed two-by-two block matrix partitioning methods and elementwise constructed preconditioners for the arising pivot block and Schur complement matrices have turned out to be very efficient methods, and are analysed in this paper. Thereby special attention is paid to macroelementwise partitionings, which can be particularly efficient in the modelling of materials with large and narrow variations and can also provide efficient implementations on parallel computers.