An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

Abstract

We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates’ transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.