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.

翻译：时谐波传播问题，特别是由麦克斯韦方程控制的问题，由于算子的非自伴性以及离散化产生的大型非厄米线性系统，带来了显著的计算挑战。区域分解方法，尤其是一层施瓦茨方法，为解决这些挑战提供了一个有前景的框架，其最新进展显示了在特定条件下实现弱可扩展性的潜力。本文分析了用于带状区域分解的麦克斯韦方程一层施瓦茨方法的弱可扩展性，重点关注具有一般横截面和不同类型传输条件（如阻抗或完全匹配层（PMLs））的波导。通过结合托普利茨矩阵极限谱分析和麦克斯韦解模态分解的技术，我们提出了一个新颖的理论框架，将先前的工作扩展到更复杂的几何结构和传输条件。数值实验证实，即使子域数量适中，极限谱也能有效预测实际行为。此外，我们证明了一层施瓦茨方法在特定的区域分解参数下能够实现关于波数的鲁棒性，这为其在大规模电磁波问题中的适用性提供了新的见解。