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

TL;DR

The paper develops a modal-based analysis of one-level Schwarz domain decomposition for Maxwell's equations in waveguides with general cross-sections, establishing weak scalability by coupling Toeplitz limiting-spectrum techniques with TE/TM/TEM mode decompositions. It shows that the Schwarz iteration decouples mode-by-mode into a block Toeplitz structure, enabling explicit spectral characterizations and a per-mode convergence dictionary that mirrors scalar Helmholtz results with a mode-wise transformation $\kappa \leftrightarrow k^2/\kappa$. Transmission conditions, including impedance and PML, diagonalize on modal traces and yield explicit per-mode symbols $\lambda_i^{\mathrm T}$, governing inter-subdomain communication. Numerical experiments validate the limiting-spectrum predictions for both impedance and PML transmissions, demonstrate rapid convergence for evanescent modes, and reveal enhanced weak scalability when absorption or PML parameters are employed; strong-sc scalability results further confirm robustness when subdomain counts grow under damped or PML-augmented settings. Together, these results provide practical guidance on when one-level Schwarz methods can be robust and scalable for large-scale electromagnetic wave simulations in complex waveguides.

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.

Figures (5)

• Figure 1: Scheme of the geometry $\Omega$. In blue, cross-sections of the waveguide.
• Figure 1: A bounded waveguide $\widetilde{\Omega}$ and its decomposition into subdomains.
• Figure 1: Spectral radius of the iteration matrix for TE (left) and TM (right) versus $r\ge 0$ with impedance transmission \ref{['def:op-T-ik']}. Dashed: limiting spectrum.
• Figure 2: Spectral radius $\rho\bigl(I^{\rm T}(r)\bigr)$ with PML transmission \ref{['def:op-PML']}. Left: $\ell=0.1,\ \sigma=5$. Right: $\ell=1,\ \sigma=10$. Dashed: limiting spectrum.
• Figure 3: Spectral radius $\rho\bigl(I^{\rm TE}(r)\bigr)$ with $k=10+\imath$ using impedance (left) and PML (right). Dashed: limiting spectrum.

Theorems & Definitions (22)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.4: Tangential traces of mode profiles
• Proof 1
• Remark 2.5
• Definition 3.1: The Schwarz algorithm
• Remark 3.2
• Proposition 3.3: Diagonalization of ${\mathbf{E}} \rightarrow ((\nabla \times {\mathbf{E}}) \times \mathbf{n}) \times \mathbf{n}$
• Proof 2