An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries
This work provides a numerically efficient and theoretically grounded approach for solving a challenging eigenvalue problem in electromagnetics, relevant to inverse scattering and non-destructive testing.
The authors develop a spectral-Galerkin method for solving Maxwell transmission eigenvalue problems in spherical geometry, reducing the problem to parallelizable one-dimensional TE and TM modes. Numerical experiments validate the theoretical error estimates and demonstrate algorithmic efficiency.
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.