Multiscale mortar mixed methods for heterogeneous elliptic problems

Consider solving a second order elliptic problem when the elliptic coefficient is highly heterogeneous. Generally, a numerical method either uses a very fine computational mesh to resolve the heterogeneities and therefore becomes computationally inefficient, or it performs efficiently on a coarse mesh but gives inaccurate results. Standard nonoverlapping domain decomposition using mortar spaces to couple together the subdomains efficiently handles these equations in parallel, but the issue of heterogeneity is not directly addressed. We define new mortar spaces that incorporate fine scale information obtained from local cell problems, using the theory of homogenization as a heuristic guide to limit the number of degrees of freedom in the mortar space. This gives computational efficiency in parallel, even when the subdomain problems are fully resolved on a fine mesh. In the case of an elliptic coefficient satisfying the two-scale separation assumption, the method is provably accurate with respect to the heterogeneity. Formally first and second order mortar space approximations are constructed explicitly in two dimensions. Numerical tests are presented for one medium with the two-scale separation assumption and two without it. The results show that these new homogenization based mortar spaces work much better than simple polynomial based mortar spaces, and that generally the second order spaces work better than the first order ones.