A Legendre–Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues

In this paper we present an efficient spectral method based on the Legendre–Galerkin approximation for the transmission eigenvalue problem. A rigorous error analysis is presented by using the minmax principle for the generalized eigenvalue problems associated to a transmission eigenvalue problem. However, this formulation can only compute real eigenvalues. Thus, we also present another formulation based on second order equations and construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. For the case of constant medium, we derive the matrix formulations based on the tensor-product for the discrete variational form in two and three-dimensional cases, respectively. In addition, we also establish an optimization scheme based on the Legendre–Galerkin approximation. With this scheme we can estimate the index of refraction of an inhomogeneous medium. We also present ample numerical results to show that our method is very effective and high accurate.