Optimized Additive Schwarz with Harmonic Extension as a Discretization of the Continuous Parallel Schwarz Method

The additive Schwarz method with harmonic extension (ASH) was introduced by Cai and Sarkis (1999) as an efficient variant of the additive Schwarz method that converges faster and requires less communication. We show that ASH can also be used with optimized transmission conditions to obtain faster convergence. We show that when the decomposition into subdomains contains no cross points, optimized ASH can be reformulated as an iteration that is closely related to the optimized Schwarz method at the continuous level. In fact, the iterates of ASH are identical to the iterates of the discretized parallel Schwarz method outside the overlap, whereas inside the overlap they are linear combinations of previous Schwarz iterates. Thus, one method converges if and only if the other one does, and they do so at the same asymptotic rate, unlike additive Schwarz, which fails to converge inside the overlap. However, when cross points are present, then ASH and the Schwarz methods are incomparable, i.e., there are cases where one method converges and the other diverges, and vice versa.