The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. The problem plays a key role in the proof of the unique determination of an inhomogeneous media. Furthermore, transmission eigenvalues can be reconstructed from the scattering data and used to estimate the material properties of the unknown object. The problem is posted as a system of two second order partial differential equations and is nonlinear and non-selfadjoint. It is challenging to develop effective numerical methods and prove the convergence. In this talk, we briefly review several finite element methods for transmission eigenvalues of isotropic media. For anisotropic media, we formulate the transmission eigenvalue problem as an eigenvalue problem of a holomorphic Fredholm operator function of index zero. The Lagrange finite elements are used for the discretization and the convergence is proved using the abstract approximation theory for holomorphic operator functions. A spectral indicator method is developed to compute the eigenvalues. Numerical examples are presented for validation.
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