Does the Partition of Unity Influence the Convergence of Schwarz Methods?

Abstract

The classical alternating Schwarz method does not need a partition of unity in its definition [3]: one solves one subdomain after the other, stores subdomain solutions, and always uses the newest data available from the neighboring subdomains. In the parallel Schwarz method introduced by Lions in [10], where all subdomains are solved simultaneously, one also stores subdomain solutions, but one has to distinguish two cases: if in the decomposition there are never more than two subdomains that intersect, which we call the no crosspoint assumption, then one also does not need a partition of unity to define the method, one simply takes data from the neighboring subdomains with which the subdomain intersects, and in that case the parallel Schwarz method has a variational interpretation [10].