Modal analysis of a domain decomposition method for Maxwell's equations in a waveguide

TL;DR

This paper analyzes the weak scalability of one-level Schwarz domain decomposition methods for Maxwell's equations in waveguides, introducing a new theoretical framework and confirming its predictive accuracy through numerical experiments.

Contribution

It extends the spectral analysis of Schwarz methods to complex geometries and transmission conditions, providing a novel theoretical approach for electromagnetic wave problems.

Findings

Limiting spectrum analysis predicts practical behavior with few subdomains.

One-level Schwarz method can be wave number robust under certain parameters.

Numerical experiments validate the theoretical predictions.

Abstract

Time-harmonic wave propagation problems, especially those governed by Maxwell's equations, pose significant computational challenges due to the non-self-adjoint nature of the operators and the large, non-Hermitian linear systems resulting from discretization. Domain decomposition methods, particularly one-level Schwarz methods, offer a promising framework to tackle these challenges, with recent advancements showing the potential for weak scalability under certain conditions. In this paper, we analyze the weak scalability of one-level Schwarz methods for Maxwell's equations in strip-wise domain decompositions, focusing on waveguides with general cross sections and different types of transmission conditions such as impedance or perfectly matched layers (PMLs). By combining techniques from the limiting spectrum analysis of Toeplitz matrices and the modal decomposition of Maxwell's solutions, we provide a novel theoretical framework that extends previous work to more complex geometries and transmission conditions. Numerical experiments confirm that the limiting spectrum effectively predicts practical behavior even with a modest number of subdomains. Furthermore, we demonstrate that the one-level Schwarz method can achieve robustness with respect to the wave number under specific domain decomposition parameters, offering new insights into its applicability for large-scale electromagnetic wave problems.