An Implementation of a Coarse-Fine Mesh Stabilized Schwarz Method for a Three-Space Dimensional PDE-Problem

When solving very large scale problems on parallel computer platforms, we consider the advantages of domain decomposition in strips or layers, compared to general domain decomposition splitting techniques. The layer sub-domains are grouped in pairs, ordered as odd-even respectively even-odd and solved by a Schwarz alternating iteration method, where the solution at the middle interfaces of the odd-even groups is used as Dirichlet boundary conditions for the even-odd ordered groups and vice versa. To stabilize the method the commonly used coarse mesh method can be replaced by a coarse-fine mesh method. A component analysis of the arising eigenvectors demonstrates that this solution framework leads to very few Schwarz iterations. The resulting coarse-fine mesh method entails a coarse mesh of a somewhat large size. In this study it is solved by two methods, a modified Cholesky factorization of the whole coarse mesh matrix and a block-diagonal preconditioner, based on the coarse mesh points and the inner node points. Extensive numerical tests show that the latter method, being also computationally cheaper, needs very few iterations, in particular when the domain has been divided in many layers and the coarse to fine mesh size ratio is not too large.