The improved interpolating element-free Galerkin method with nonsingular weight functions for 3D Schrödinger equations

Abstract

In this study, the improved interpolating element-free Galerkin (IIEFG) method with a nonsingular weight function for solving the 3D Schrödinger equations is presented. We apply the improved interpolating moving least-squares (IIMLS) method with non-singular weight functions to form shape functions. The shape functions of the IIMLS method satisfy the properties of the Kronecker δ-functions and allow the direct imposition of essential boundary conditions. The IIMLS approach can successfully address the MLS method's issues caused by the weighting function's singularity. In comparison to the MLS approximation approach, the IIMLS approach incorporates less unknown coefficients in its test function. Thus, in this article, the IIEFG approach is used to solve the 3D Schrödinger equations by combining the IIMLS approach with the Galerkin weak form. The IIEFG approach in this paper offers greater efficiency in terms of computational accuracy and computational efficiency compared to the improved element-free Galerkin (IEFG) method. To illustrate the superiority of the IIEFG method, three numerical examples are solved by the IIEFG method. The 3D Schrödinger equation is a complex-variable equation, and this study employs a real-virtual partially discretized method to solve its numerical solution, which provides a new idea for the subsequent study of complex-variable equations by meshless methods.