The Efficient Element-Free Galerkin Method for 3D Schrödinger Equations

This study proposes the efficient element-free Galerkin (EEFG) method, which is based on the improved moving least-squares (IMLS) method, for addressing the three-dimensional (3D) Schrödinger equations. The dimension splitting method (DSM) is vital to this approach, since it splits the 3D Schrödinger equation in a specific direction and transforms it into a sequence of two-dimensional (2D) and one-dimensional (1D) issues. The improved element-free Galerkin (IEFG) method is subsequently applied to solve the corresponding 2D problems, using the finite difference method (FDM) employed for both the splitting direction and the time domain. Finally, the EEFG method for addressing the 3D Schrödinger equations is established. Compared to the IEFG approach, the EEFG method offers greater computational accuracy. Furthermore, this method outperforms the IEFG method in terms of computational efficiency for the same precision. To demonstrate the problem, numerical instances are solved using the EEFG approach, demonstrating its advantages in terms of computing correctness and efficiency.