Is Additive Schwarz with Harmonic Extension Just Lions’ Method in Disguise?

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 how ASH, which is defined at the matrix level, can be reformulated as an iteration that bears a close resemblance to the parallel Schwarz method at the continuous level, provided that the decomposition of subdomains contains no cross points. 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, the two methods converge with the same asymptotic rate, unlike additive Schwarz, which fails to converge inside the overlap (Efstathiou & Gander 2007).