A Weak Galerkin Finite Element Method Coupled with Mortar Spectral Element Method for Schrödinger Eigenvalue Problem with an Inverse Square Potential

Abstract

In this paper, we introduce a weak Galerkin (WG) finite element method coupled with mortar spectral element method (MSEM) to solve the Schrödinger eigen-value problem with an inverse square potential. For the domain around the inverse square potential, we use the mortar spectral element method to simulate the singularities in eigenfunctions caused by the inverse square potential, while we employ the WG method in the remaining domain. This coupled method can effectively handle the singularity arising from the inverse square potential. Notably, hanging nodes are allowed on the coupled interface. Compared to the conforming finite element method coupled with MSEM, our approach is not constrained by the mesh size of the mortar spectral element. This flexibility permits the use of fine meshes in the WG domain, thereby enhancing accuracy. We provide hp error analysis for both eigenfunctions and eigenvalues. Numerical experiments demonstrate the hp convergence of the theoretical results.