Mixed spectral-element method for 3-D Maxwell's eigenvalue problem

A high-order 3-D mixed spectral-element method (SEM) based on Gauss-Lobatto-Legendre polynomials in the mixed finite-element framework is proposed to remove the spurious eigenmodes in the numerical solution of the vector Maxwell eigenvalue problem with inhomogeneous, lossy isotropic, and anisotropic media. In order to suppress all zero and nonzero spurious modes that exist in the conventional finite-element and higher order SEMs, the proposed method not only employs the mixed-order curl-conforming vector basis functions for the electric field intensity, but also includes the divergence-free condition given by Gauss' law in a weak form. Several numerical examples are given to verify that the mixed SEM is free of any spurious eigenmodes and has spectral accuracy with analytic eigenvectors.