On the Design of Small Coarse Spaces for Domain Decomposition Algorithms

Abstract

Methods are presented for automatically constructing coarse spaces of low dimension for domain decomposition algorithms. 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 such as METIS; 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 are well suited for use in two-level or multilevel preconditioners such as overlapping Schwarz algorithms. An analysis for scalar elliptic and linear elasticity problems reveals that significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. These estimates depend primarily on the Lipschitz parameters of the subdomains. Numerical examples f...