Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

In this paper, we study domain decomposition preconditioners for multiscale flows in high contrast media. Our problems are motivated by porous media applications where low conductivity regions play an important role in determining flow patterns. We consider flow equations governed by elliptic equations in heterogeneous media with large contrast between high and low conductivity regions. This contrast brings an additional small scale (in addition to small spatial scales) into the problem expressed as the ratio between low and high conductivity values. Using weighted coarse interpolation, we show that the condition number of the preconditioned systems using domain decomposition methods is independent of the contrast. For this purpose, Poincaré inequalities for weighted norms are proved in the paper. The results are further generalized by employing extension theorems from homogenization theory. Our numerical observations confirm the theoretical results.