发表机构
Southeast University; Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); National Yang Ming Chiao Tung University; Nanjing Normal University（东南大学; 上海数学与交叉学科研究院; 国立阳明交通大学; 南京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种通过六维嵌入求解三维准周期Maxwell本征问题的数值框架，利用显式正交基消除零空间，采用逆Lanczos方法及局部Rayleigh商验证，实现高效准确的模式恢复。
AI 中文摘要
我们开发了一个数值框架，用于通过六维周期嵌入求解三维准周期Maxwell本征值问题。投影Bloch-Fourier离散化产生一个具有大型梯度型核的结构化广义本征值问题。纵向和横向子空间的显式正交基精确地移除了该核，并将原始广义本征值问题简化为仅包含正谱的无零空间标准本征值问题。约化算子的显式逆表示避免了嵌套的内-外线性求解，并导致一种逆Lanczos方法，其主要的内部计算是Hermitian正定共轭梯度求解，条件数受质量矩阵条件数限制；残差估计量化了内部求解对逆Ritz对的影响。为了在物理空间中恢复计算模式，六维Fourier特征向量通过分离的多中心Taylor展开在三维Yee网格上重建，这避免了密集的Fourier到网格相位矩阵，并在直接密集重建变得过于昂贵时保持实用。局部加权Rayleigh商提供了独立的物理空间验证，并证明了在单位分割条件下，其质量加权期望等于裁剪后的Yee Rayleigh商。数值实验确认了所提出框架的准确性和计算效率，以及其对三维准周期Maxwell模式的物理空间恢复。
英文摘要
We develop a numerical framework for three-dimensional quasiperiodic Maxwell eigenvalue problems obtained through a six-dimensional periodic embedding. A projected Bloch--Fourier discretization yields a structured generalized eigenvalue problem with a large gradient-type kernel. Explicit orthonormal bases for the longitudinal and transverse subspaces remove this kernel exactly and reduce the original generalized eigenvalue problem to a null-space-free standard eigenvalue problem containing only the positive spectrum. An explicit inverse representation of the reduced operator avoids nested inner--outer linear solves and leads to an inverse Lanczos method whose main inner computation is a Hermitian positive definite conjugate-gradient solve with condition number bounded by that of the mass matrix; a residual estimate quantifies the effect of inner solves on the inverse Ritz pairs. To recover the computed modes in physical space, the six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional Yee grid by a separated multi-center Taylor expansion, which avoids the dense Fourier-to-grid phase matrix and remains practical when direct dense reconstruction becomes prohibitively expensive. Local weighted Rayleigh quotients provide an independent physical-space validation, and their mass-weighted expectation is proved to equal the cropped Yee Rayleigh quotient under a partition-of-unity condition. Numerical experiments confirm the accuracy and computational effectiveness of the proposed framework and its physical-space recovery of three-dimensional quasiperiodic Maxwell modes.
