Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over the last decade, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. We show here why the classical Schwarz method applied to the hyperbolic problem converges without overlap for Maxwell's equations. The reason is that the method is equivalent to a simple optimized Schwarz method for an equivalent elliptic problem. Using this link, we show how to develop more efficient Schwarz methods than the classical ones for the Maxwell's equations. We illustrate our findings with numerical results.
Optimized Schwarz preconditioning for SEM based magnetohydrodynamics
Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research（国家大气研究中心应用数学与地球科学研究所）；Department of Mathematics, Kyungpook National University（庆北大学数学系）
Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell's equations
Université Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；LJAD（约瑟夫·拉格朗日实验室）；University of Strathclyde（斯特拉斯克莱德大学）；UPMC Univ Paris 6（巴黎第六大学皮埃尔和玛丽·居里大学）；LJLL（拉格朗日分析与偏微分方程实验室）
New optimized Schwarz algorithms for one dimensional Schrödinger equation with general potential
Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis（雅克·阿达马·迪厄多内数学实验室，法国国家科学研究中心联合研究单位7351，尼斯索菲亚抗极地大学）；Team COFFEE, INRIA Sophia Antipolis Méditerranée（COFFEE团队，尼斯索菲亚抗极地地中海国家信息与自动化研究所）；BRGM Orléans France（法国奥尔良地球科学研究局）
A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods
University of Geneva（日内瓦大学）；Zhejiang Ocean University（浙江海洋大学）
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第七大学SPC，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné（蔚蓝海岸大学，法国国家科学研究中心，J-A 迪厄多内实验室）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）
An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations
Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources
