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

TL;DR

This paper investigates how relaxation and acceleration techniques, specifically Aitken and Anderson accelerations, influence the convergence of the non-overlapping Schwarz method in domain decomposition coupling, proposing adaptive variants and comparing their effectiveness.

Contribution

The paper introduces an adaptive Anderson acceleration method and compares it with Aitken acceleration, providing insights into their effectiveness for different domain decomposition scenarios.

Findings

Aitken acceleration is most efficient for two sub-domain problems.

Anderson acceleration performs better in multi-domain settings.

The adaptive Anderson method improves convergence robustness.

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.