Small Coarse Spaces for Overlapping Schwarz Algorithms with Irregular Subdomains

Abstract

Methods are developed for automatically constructing small coarse spaces of low dimension for domain decomposition algorithms for problems in three dimensions. These constructions use equivalence classes of nodes on the interface between the subdomains into which the domain of a given elliptic problem has been subdivided, e.g., by a mesh partitioner; these equivalence classes already play a central role in the design, analysis, and programming of many domain decomposition algorithms. The coarse space elements are well defined even for irregular subdomains, are continuous, and well suited for use in two-level or multi-level preconditioners such as overlapping Schwarz algorithms. Significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. The condition number estimates depend primarily on the Lipschitz parameters of the subdomains.