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

Author

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  • Jianyun Wang

    (School of Science, Hunan University of Technology, Zhuzhou 412007, China
    School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China)

  • Zixin Zhong

    (School of Science, Hunan University of Technology, Zhuzhou 412007, China)

  • Zhikun Tian

    (School of Computational Science and Electronics, Hunan Institute of Engineering, Xiangtan 411104, China)

  • Ying Liu

    (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 H 1 -norm and carry out two numerical experiments.

Suggested Citation

  • Jianyun Wang & Zixin Zhong & Zhikun Tian & Ying Liu, 2024. "A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation," Mathematics, MDPI, vol. 12(5), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:726-:d:1348672
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