Finite element/holomorphic operator function method for the transmission eigenvalue problem

The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. It plays a key role in the unique determination of 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-self adjoint. It is challenging to develop effective numerical methods. In this paper, we formulate the transmission eigenvalue problem as the eigenvalue problem of a holomorphic operator function. The Lagrange finite elements are used for the discretization and the convergence is proved using the abstract approximation theory for holomorphic Fredholm operator functions. The spectral indicator method is employed to compute the eigenvalues. Numerical examples are presented to validate the proposed method