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

## TL;DR

This paper develops an efficient spectral-Galerkin method for Maxwell transmission eigenvalue problems in spherical geometries, utilizing vector spherical harmonics and specialized algorithms for TE and TM modes, with proven error estimates and validated numerical results.

## Contribution

It introduces a novel spectral-Galerkin approach with error analysis for Maxwell transmission eigenvalues in spherical geometries, including algorithms for TE and TM modes.

## Key findings

- Spectral-Galerkin method achieves high accuracy for eigenvalues.
- Parallelizable solution process for TE and TM modes.
- Numerical experiments confirm theoretical error estimates.

## Abstract

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