It is well known tha the conventional edge element method in solving vector Maxwell's eigenvalue problem will lead to the presence of nonphysical zero eigenvalues. This paper uses the mixed finite element method to suppress the presence of these nonphysical zero eigenvalues for 2D vector Maxwell's eigenvalue problem in anisotropic media. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on employing the first-order edge element basis functions to expand the electric field and linear nodal element basis functions to expand the Lagrangian multiplier. Our numerical experiments show that this method can successfully remove all nonphysical zero and nonzero eigenvalues. We verify that when the cavity has a connected perfect electric boundary, then there is no physical zero eigenvalue. Otherwise, the number of physical zero eigenvalues is one less than the number of disconnected perfect electric boundaries.
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