Numerische Mathematik Manuscript-nr. Overlapping Schwarz Methods on Unstructured Meshes Using Non-matching Coarse Grids

We consider two level overlapping Schwarz domain decomposition methods for solving the nite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard nite element interpolation from the coarse to the ne grid may be used. Our theory requires no assumption on the substructures that constitute the whole domain, so the substructures can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the ne grid on which the discrete problem is to be solved, and neither the coarse mesh nor the ne mesh need be quasi-uniform. In addition, the domains deened by the ne and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the ne grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes. The condition number of the preconditioned system depends only on the (possibly small) overlap of the substructures and the size of the coarse grid, but is independent of the sizes of the subdomains.