An Immersed Finite Element Approach Based on Dimensional Reduction for Elliptic Interface Eigenvalue Problems with Radial Stratified Media

In this paper, a novel and efficient immersed finite element approach based on dimensional reduction is proposed and studied for elliptic interface eigenvalue problems with radial stratified media. First, by introducing an appropriate coordinate transformation and leveraging the orthogonal properties of Fourier series and spherical harmonics, the original high-dimensional problem is decomposed into a series of decoupled one-dimensional second-order interfacial eigenvalue problems. Subsequently, a class of weighted Sobolev spaces and their approximation spaces are defined in accordance with the boundary conditions and interface jump conditions, based on which, the variational form and its discrete scheme are established for each one-dimensional second-order interface eigenvalue problem. Finally, a rigorous theoretical proof is provided to demonstrate the error estimates for approximate eigenvalues and eigenfunctions. The error order for approximate eigenvalues is Formula: see text, while that for eigenfunctions is Formula: see text. Numerical experiments validate the correctness of the theoretical analysis and confirm the convergence performance of the algorithm.