  This paper mainly investigates the classic resonant cavity problem with
anisotropic and nonconductive media, which is a linear vector Maxwell's
eigenvalue problem. The finite element method based on edge element of the
lowest-order and standard linear element is used to solve this type of 3-D
closed cavity problem. In order to eliminate spurious zero modes in the
numerical simulation, the divergence-free condition supported by Gauss' law is
enforced in a weak sense. After the finite element discretization, the
generalized eigenvalue problem with a linear constraint condition needs to be
solved. Penalty method, augmented method and projection method are applied to
solve this difficult problem in numerical linear algebra. The advantages and
disadvantages of these three computational methods are also given in this
paper. Furthermore, we prove that the augmented method is free of spurious
modes as long as the anisotropic material is not magnetic lossy. The projection
method based on singular value decomposition technique can be used to solve the
resonant cavity problem. Moreover, the projection method {cannot} introduce any
spurious modes. At last, several numerical experiments are carried out to
verify our theoretical results.
