Efficient domain decomposition methods for elliptic problems arising from flows in heterogeneous porous media

Abstract

In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition methods for these problems requires a coarse solver which is adaptive to the fine scale features, [4]. We propose the use of a multiscale coarse solver based on a finite volume – finite element formulation. The resulting domain decomposition methods seem to induce a convergence rate nearly independent of the aspect ratio of the extreme permeability values within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the preconditioner through numerical experiments which include problems with multiple scale coefficients, as well as problems with continuous scales.