An efficient decoupled and dimension reduction scheme for quad-curl eigenvalue problem in balls and spherical shells

In this paper, we propose a spectral-Galerkin approximation for the quad-curl eigenvalue problem within spherical geometries. Utilizing vector spherical harmonics in conjunction with the Laplace-Beltrami operator, we decompose the quad-curl eigenvalue problem into two distinct categories of fourth-order equations: corresponding to the transverse electric (TE) and transverse magnetic (TM) modes. A thorough analysis is provided for the TE mode. The TM mode, however, is characterized by a system of coupled fourth-order equations that are subject to a divergence-free condition. We develop two separate sets of vector basis functions tailored for the coupled system in both solid spheres and spherical shells. Moreover, we design a parameterized technique aimed at eliminating spurious eigenpairs. Numerical examples are presented to demonstrate the high precision achieved by the proposed method. We also include graphs to illustrate the localization of the eigenfunctions. Furthermore, we employ Bessel functions to analyze the quad-curl problem, revealing the intrinsic connection between the eigenvalues and the zeros of combinations of Bessel functions.