SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1039-1066, June 2024.
Abstract. In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm.

中文翻译：

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麦克斯韦传输特征值问题的一种新型混合谱方法和误差估计

《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1039-1066 页，2024 年 6 月
。摘要。本文针对球域中的麦克斯韦传输特征值问题，提出并研究了一种新颖的混合谱伽辽金方法。该方法利用矢量球谐函数来实现降维。通过引入辅助向量函数，将原问题重写为等效四阶耦合线性本征系统，并进一步分解为一维四阶解耦横电（TE）和横磁（TM）序列模式。基于紧嵌入理论和紧算子的谱逼近性质，建立了TE模式的特征值和特征函数逼近的误差估计。对于TM模式，设计了高效的本质极性条件和高阶多项式逼近方法来应对耦合边界条件引起的奇异性和复杂性。数值实验证明了我们算法的效率和鲁棒性。