Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios

Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a classof elliptic problems arising from composite materials and flows in porous media which contain many spatialscales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarsesolver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domaindecomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate inthe presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent ofthe aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework iscarried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numericalexperiments which include problems with multipl