Mass conservative domain decomposition preconditioners for multiscale finite volume method

Abstract

In this paper, we propose some new coarse correction matrices to design domain decomposition (DD) preconditioners for solving a multiscale finite volume algebraic system. The key ingredients of our coarse correction matrices are based on several operators: prolongation, restriction, and correction operators. Using the coarse correction matrices, together with a one-level additive Schwarz method as a local solver, we may get several efficient preconditioners for the fine-scale finite volume linear system. Techniques used in some well-known multiscale methods are important and inspire us to combine them to generate different kinds of operators. It is shown that our new coarse correction matrices are more robust than well-known ones in the literature. In addition, mass conservation on a primal coarse grid is preserved by a postprocessing procedure, so a conservative fine velocity field may be reconstructed in any interesting local domain. A variety of numerical examples are presented to confirm the validity and robustness of our coarse correction matrices.