Title: Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions


Authors: Clark R. Dohrmann, Axel Klawonn, and Olof B. Widlund

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 accommodates subdomains with much less regular shape are presented; the subdomains 
are only required 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 
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 subregions including those obtained when a mesh partitioner is 
used for the domain decomposition.