  The transmission eigenvalue problem arises from the inverse scattering theory
for inhomogeneous media and has important applications in many qualitative
methods. The problem is posted as a system of two second order partial
differential equations and is essentially nonlinear, non-selfadjoint, and of
higher order. It is nontrivial to develop effective numerical methods and the
proof of convergence is challenging. In this paper, we formulate the
transmission eigenvalue problem for anisotropic media as an eigenvalue problem
of a holomorphic Fredholm operator function of index zero. The Lagrange finite
elements are used for 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.
