On the role of relaxation and acceleration in the non-overlapping Schwarz alternating method for coupling

Abstract: The purpose of this paper is to study the influence of relaxation and acceleration techniques on the convergence behavior of the non-overlapping Schwarz algorithm with alternating Dirichlet-Neumann transmission conditions in the context of domain decomposition- (DD-) based coupling. After demonstrating that the multiplicative Schwarz scheme can be formulated as a fixed-point iteration, we explore, both theoretically and numerically, two promising techniques for speeding up the method: (i) Aitken acceleration and (ii) Anderson acceleration. In the process, we derive a robust and efficient adaptive variant of Anderson acceleration, termed "Anderson with memory adaptation". We compare the proposed acceleration strategies to the well-known classical relaxed Dirichlet-Neumann Schwarz alternating method. Our results suggest that, while Aitken-accelerated Schwarz is the best approach in terms efficiency and robustness when considering two sub-domain DDs, Anderson-accelerated Schwarz is the method of choice in larger multi-domain setting.