A convergence theory for an overlapping Schwarz algorithm using discontinuous iterates

Abstract

A new type of overlapping Schwarz methods, using discontinuous iterates, is constructed by modifying the classical overlapping Schwarz algorithm. This new algorithm allows for discontinuous iterates across the artificial interface. For Poisson’s equation, this algorithm can be considered as an overlapping version of Lions’ Robin iteration method for which little is known concerning the rate of convergence. Since overlap improves the performance of the classical algorithms considerably, the existence of a uniform convergence factor is the fundamental question for our algorithm. A new theory using Lagrange multipliers is developed and conditions are found for the existence of an almost uniform convergence factor for the dual variables, which implies rapid convergence of the primal variables, in the two overlapping subdomain case. Our result also shows a relation between the boundary conditions of the given problem and the artificial interface condition. Numerical results for the general case with cross points are also presented. They indicate possible extensions of our results to this more general case.