Iterative Methods for Transmission Eigenvalues

Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of the physical properties, such as the index of refraction, of the scattering target. Despite considerable effort devoted to the existence and estimation for the transmission eigenvalues, the numerical treatment is limited. Since the problem is non-standard, classical finite element methods result in non-Hermitian matrix eigenvalue problems. In this paper, we focus on the computation of a few lowest transmission eigenvalues which are of practical importance. Instead of a non-Hermitian problem, we work on a series of generalized Hermitian problems. We first use a forth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem. The roots of these functions are the transmission eigenvalues. Then we apply iterative methods to compute the transmission eigenvalues. We show the convergence of the numerical schemes. The effectiveness of the methods is demonstrated using various numerical examples.