Multiscale finite element and domain decomposition methods for high-contrast problems using local spectral basis functions

In this paper, we study multiscale finite element methods (MsFEMs) using basis functions that provide an optimal convergence in domain decomposition methods. We consider second order elliptic equations with highly variable coefficients. Both multiscale finite element and domain decomposition methods (considered here) use coarse spaces to achieve efficiency and robustness. In MsFEMs, the spatial variability of the media affects the convergence rate (see e.g., [14]). In domain decomposition methods, high variability of the coefficients within coarse regions affects the number of iterations required for the convergence. In this paper, we use coarse spaces designed for high-contrast problems that provide an optimal convergence in domain decomposition methods. These spaces are constructed locally. In particular, basis functions are constructed using solutions of a local spectral problem. In our previous work [18], we show that using these coarse spaces one can construct preconditioners such that the condition number of the preconditioned system is independent of the contrast. In this paper, these coarse spaces are used in MsFEMs to solve elliptic equations with high-contrast heterogeneous coefficients on a coarse grid. Our numerical results show that MsFEMs with coarse spaces constructed via local spectral problems are more accurate compared to multiscale methods that employ traditional multiscale spaces, e.g., with linear boundary conditions. However, we argue that these coarse spaces based on local eigenvalue problems are not sufficient to capture the fine-scale behavior of the solution very accurately. In particular, these coarse spaces identify high conducting regions accurately, but they may not capture the detailed behavior of the solution in other regions, such as between high-conductivity regions. The latter can be important for the ∗Department of Mathematics, Texas A & M University, College Station, TX 77843 ([email protected]) †Department of Mathematics, Texas A & M University, College Station, TX 77843 ([email protected]) ‡ExxonMobil Upstream Research Company, P.O. Box 2189, Houston, TX 77252 ([email protected])