A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation-Reference-Cited by-同舟云学术

A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

Published:2024-02-29 Issue:5 Volume:12 Page:726
ISSN:2227-7390
Container-title:Mathematics
language:en
Short-container-title:Mathematics

Author:

Wang Jianyun¹²,Zhong Zixin¹,Tian Zhikun³,Liu Ying⁴

Affiliation:

1. School of Science, Hunan University of Technology, Zhuzhou 412007, China

2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China

3. School of Computational Science and Electronics, Hunan Institute of Engineering, Xiangtan 411104, China

4. College of Information and Intelligence, Hunan Agricultural University, Changsha 410128, China

Abstract

In this paper, we construct a new two-grid algorithm of the finite element method for the Schrödinger equation in backward Euler and Crank–Nicolson fully discrete schemes. On the coarser grid, we solve coupled real and imaginary parts of the Schrödinger equation. On the fine grid, real and imaginary parts of the Schrödinger equation are decoupled, and we solve the elliptic equation about real and imaginary parts, respectively. Then, we obtain error estimates of the exact solution with the two-grid solution in the H1-norm and carry out two numerical experiments.

Funder

National Natural Science Foundation of China

Publisher

MDPI AG

Link

https://www.mdpi.com/2227-7390/12/5/726/pdf

Reference26 articles.

1. On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation;Akrivis;Numer. Math.,1991

2. Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain;Antonopoulou;Math. Comp.,2015

3. A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations;Li;J. Comput. Phys.,2018

4. Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation;Shi;Appl. Math. Lett.,2014

5. Superconvergence analysis of bilinear finite element for the nonlinear Schrödinger equation on the rectangular mesh;Tian;Adv. Appl. Math. Mech.,2018