Two‐Level Schwarz Algorithms with Overlapping Subregions for Mortar Finite Elements

Abstract

Preconditioned conjugate gradient methods based on two-level overlapping Schwarz methods often perform quite well. Such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined. Precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases. In this paper, this domain decomposition algorithm and analysis are extended to mortar finite elements. It is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered.