On Iterative Substructuring Methods for Multiscale Problems

In this note, we discuss iterative substructuring methods for a scalar elliptic model problem with a strongly varying diffusion coefficient that is typically discontinuous and exhibits large jumps. Opposed to earlier theory, we treat the case where the jumps happen on a small spatial scale and can in general not be resolved by a domain decomposition. We review the available theory of FETI methods for coefficients that are—on each subdomain (or a part of it)—quasi-monotone. Furthermore, we present novel theoretical robustness results of FETI methods for coefficients which have a large number of inclusions with large values, and a constant “background” value (by far not quasi-monotone). In both cases, the coarse space is the usual space of constants per subdomain.