Cheng Xiaoliang, Yang Jing
J Opt Soc Am A Opt Image Sci Vis. 2015 Jul 1;32(7):1243-6. doi: 10.1364/JOSAA.32.001243.
We consider a computational method for the interior transmission eigenvalue problem that arises in acoustic and electromagnetic scattering. The transmission eigenvalues contain useful information about some physical properties, such as the index of refraction. Instead of the existence and estimation of the spectral property of the transmission eigenvalues, we focus on the numerical calculation, especially for spherically stratified media in R. Due to the nonlinearity and the special structure of the interior transmission eigenvalue problem, there are not many numerical methods to date. First, we reduce the problem into a second-order ordinary differential equation. Then, we apply the Hermite finite element to the weak formulation of the equation. With proper rewriting of the matrix-vector form, we change the original nonlinear eigenvalue problem into a quadratic eigenvalue problem, which can be written as a linear system and solved by the eigs function in MATLAB. This numerical method is fast, effective, and can calculate as many transmission eigenvalues as needed at a time.
