An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries

Abstract

In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.