Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions

In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by C(1 + H/δ)(1 + log(H/h))², where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subrogions including those obtained when a mesh partitioner is used for the domain decomposition.