An Effective Finite Element Method for Maxwell Eigenvalue Problem in a Radial Inhomogeneous Medium

In this paper, an effective finite element method is developed and studied for the Maxwell eigenvalue problem with a radial inhomogeneous medium in a ball. The primary idea is to expand the electric field using vector spherical harmonics, and further decompose the original problem into two types of decoupled one-dimensional eigenvalue problems, namely the TE and TM modes, through their orthogonality. For the TE mode, we initially establish the abstract spectral approximation results and then give error estimates for numerical eigenvalues and eigenfunctions, leveraging the approximation properties of interpolation operators. For the TM mode, we introduce the Sobolev product space, establish its variational form and finite element discretization, design a set of vector basis functions, and derive the equivalent matrix form of the discretization. Finally, we present numerous numerical examples, and the numerical results demonstrate the stability and effectiveness of our algorithm.