A study of the effects of irregular subdomain boundaries on some domain decomposition algorithms

In domain decomposition we decompose a global domain into subdomains. The shape and form of these subdomains may have an effect upon the overall performance of our methods. In this paper we study the effects of irregular subdomain boundaries on some domain decomposition algorithms. We reintroduce the additive average Schwarz method. This method is very flexible with respect to subdomain shapes, and we make the claim that for the scalar elliptic equation, the established convergences estimate for this method also hold under the weak conditions that subdomains are John domains with a uniformly bounded John constant. We present some numerical results to show how various subdomain shapes and geometries can effect the condition number of the preconditioned system.