Null--Space--Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems
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.