Title: An analysis of a FETI--DP algorithm on irregular subdomains in the plane


(NYU-CS-TR889)

Authors: Axel Klawonn, Oliver Rheinbach, and Olof B. Widlund

Abstract:
In the theory for domain decomposition algorithms of the iterative substructuring 
family, each subdomain is typically assumed to be the union of a few coarse triangles 
or tetrahedra. This is an unrealistic assumption, in particular, if the subdomains 
result from the use of a mesh partitioner in which case they might not even have
uniformly Lipschitz continuous boundaries. 

The purpose of this study is to derive bounds for the condition number of these  
preconditioned conjugate gradient methods which depend only on a parameter in an 
isoperimetric inequality and two geometric parameters characterizing John and uniform 
domains. A related purpose is to explore to what extent well known technical tools 
previously developed for quite regular subdomains can be extended to much more irregular
subdomains.

Some of these results are valid for any John domains, while an extension theorem, which 
is needed in this study, requires that the subdomains are uniform. The results, so far, 
are only complete for problems in two dimensions. Details are  worked out for a FETI--DP 
algorithm and numerical results support the findings. Some of the numerical experiments 
illustrate that care must be taken when selecting the scaling of the preconditioners in 
the case of irregular subdomains.