On the role of relaxation and acceleration in the non-overlapping Schwarz alternating method for coupling
This work addresses convergence improvements for domain decomposition methods, which are incremental optimizations for computational simulations in fields like engineering and physics.
The paper studied how relaxation and acceleration techniques affect the convergence of the non-overlapping Schwarz alternating method in domain decomposition coupling, finding that Aitken acceleration is best for two sub-domains while Anderson acceleration excels in multi-domain settings.
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.