The microwave resonator considered in this paper is not only filled with some piecewise homogeneous, lossless, and anisotropic media, but also has a complex geometric topology. This resonant cavity may have several physical dc eigenmodes if the geometry region occupied by the cavity is not homeomorphic to the sphere in 3-D Euclidean space. We point out that the nonzero physical eigenmodes simulated by the original source-free Maxwell's equations of first order and a system consisting of a second-order vector wave equation, a scalar equation constrained by Gauss's law and boundary conditions are equivalent, but the physical dc eigenmodes simulated by these systems are not equivalent. Furthermore, the governing equations with magnetic field as working variable is successfully solved by mixed finite-element method, which can eliminate all the nonphysical modes, including nonphysical dc modes. Finally, four numerical experiments are carried out to test the correctness of our theory.