Coarse-Grid Multiscale Model Reduction Techniques for Flows in Heterogeneous Media and Applications

In this paper, we give an overview of our results [36, 39, 46, 47] from the point of view of coarse-grid multiscale model reduction by highlighting some common issues in coarse-scale approximations and two-level preconditioners. Reduced models discussed in this paper rely on coarse-grid spaces computed by solving local spectral problems. We define local spectral problems with a weight function computed with a choice of initial multiscale basis functions. We emphasize the importance of this initial choice of multiscale basis functions for both coarse-scale approximation and for preconditioners. In particular, we discuss various choices of initial basis functions and use some of them in our simulations. We show that a naive choice of initial basis functions, e.g., piecewise linear functions, can lead to a large dimensional spaces that are needed to achieve (1) a reasonable accuracy in the coarse-scale approximation or (2) contrast-independent condition number of preconditioned matrix within two-level additive Schwarz methods. While using a careful choice of initial spaces, we can achieve (1) and (2) with smaller dimensional coarse spaces.