A Two-Level Schwarz Preconditioner for Heterogeneous Problems

Coarse space correction is essential to achieve algorithmic scalability in domain decomposition methods. Our goal here is to build a robust coarse space for Schwarz–type preconditioners for elliptic problems with highly heterogeneous coefficients when the discontinuities are not just across but also along subdomain interfaces, where classical results break down [3, 6, 15, 9]. In previous work, [7], we proposed the construction of a coarse subspace based on the low-frequency modes associated with the Dirichlet-to-Neumann (DtN) map on each subdomain. A rigorous analysis was recently provided in [2]. Similar ideas to build stable coarse spaces, based on the solution of local eigenvalue problems on entire subdomains, can be found in [4], and even traced back to similar ideas for algebraic multigrid methods in [1]. However, we will argue below that the DtN coarse space presented here is better designed to deal with coefficient variations that are strictly interior to the subdomain, being as robust as, but leading to a smaller dimension than the coarse space analysed in [4]. The robustness result that we obtain, generalizes the classical estimates for overlapping Schwarz methods to the case where the coarse space is richer than just the constant mode per domain [8], or other classical coarse spaces (cf. [15]). The analysis is inspired by that in [4, 13] and crucially uses the framework of weighted Poincaré inequalities, introduced in [12, 10] and successfully applied also to other methods in [11, 14].