An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries

摘要

We propose and analyze an efficient spectral-Galerkin approximation for theMaxwell transmission eigenvalue problem in spherical geometry. Using a vectorspherical harmonic expansion, we reduce the problem to a sequence of equivalentone-dimensional TE and TM modes that can be solved individually in parallel.For the TE mode, we derive associated generalized eigenvalue problems andcorresponding pole conditions. Then we introduce weighted Sobolev spaces basedon the pole condition and prove error estimates for the generalized eigenvalueproblem. The TM mode is a coupled system with four unknown functions, which ischallenging for numerical calculation. To handle it, we design an effectivealgorithm using Legendre-type vector basis functions. Finally, we provide somenumerical experiments to validate our theoretical results and demonstrate theefficiency of the algorithms.