The Maxwell’s eigenvalue problem is an important part of computational electromagnetics. In this report, the mixed spectral element method (Mixed SEM) by employing the Kikuchi’s mixed weak form which includes the divergence-free condition given by Gauss’ law to solve the two-dimensional and three-dimensional vector Maxwell’s eigenvalue problem with inhomogeneous, lossy isotropic, and anisotropic media. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials to construct the vector curl-conforming basis functions for the electric field and the completely continuous nodal basis functions for the auxiliary variable. Meanwhile, when solving electrically small problems similar to the low-frequency (or fine discretization) case in which the element size is far smaller than the wavelength, low-frequency breakdown occurs in the traditional full-wave simulation algorithms, which shows that the system matrix is ill-conditioned and results in slow convergence in an iterative solver or renders the inaccurate solution of a director solver. Then, to solve the low-frequency breakdown problem, the mixed SEM is extended to the time-domain. The full-wave electromagnetic (EM) field simulation algorithm by mixed SETD (MSETD) is employed. Numerical results show that the condition number of the system matrix of MSETD methods is always very small in the simulation from high frequency to DC, which eliminates the singularity of the system matrix and fundamentally overcomes the low-frequency breakdown problem.