An efficient spectral Galerkin approximation to a Helmholtz transmission eigenvalue problem in spherical geometries

Abstract

In this paper, we present an efficient spectral method based on Legendre-Galerkin approximation for the Helmholtz transmission eigenvalue problem in spherical geometries. By means of the spherical coordinate transformation and spherical harmonic expansion, we decompose the original problem into a sequence of equivalent one-dimensional generalized eigenvalue problems. Then we establish corresponding weak forms and discrete schemes for each one-dimensional generalized eigenvalue problem. Especially for the case of solid spherical domain, we need to deal with singularities introduced by spherical coordinate transformation. In order to overcome the difficulty, we derive the pole conditions and introduce the weighted Sobolev space according to pole conditions. The corresponding weak forms and discrete schemes are also established according to the weighted Sobolev space. In addition, we prove the error estimates of eigenvalues and eigenfunctions by using the spectral theory of compact operators for each one-dimensional generalized eigenvalue problems.We also provide some numerical experiments to demonstrate the validity of the algorithm and the correctness of the theoretical results.