Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Optimized Schwarz methods are based on transmission conditions between subdomains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but were almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the optimization, and we obtain four interesting new results: first, we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates. Second, we develop an asymptotically accurate approximation technique, based on Turán type inequalities in our case, to solve the much harder optimization problem on the curved interface, and this approximation technique will also be applicable to currently too complex best approximation problems in the area of optimized Schwarz methods. Third, we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized Schwarz method. Finally, we find that in the case of curved interfaces, optimized Schwarz methods are not necessarily convergent for all admissible parameters. Our optimization leads, however, to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition. We illustrate our analysis with numerical experiments.