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.