A Convergence Theory of Multilevel Additive Schwarz Methods on Unstructured Meshes

. We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, they need not be related to one another, and the domains formed by the coarse and fine grids may not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for non-selfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusion..