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Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation

Author

Listed:
  • Zhikun Tian

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

  • Yanping Chen

    (School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China)

  • Jianyun Wang

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

Abstract

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L 2 -norm with order O ( τ + h k + 1 ) . Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L 2 -norm with order O ( τ + h k + 1 + H k + 2 ) . The numerical experiment is provided to demonstrate the efficiency of the algorithm.

Suggested Citation

  • Zhikun Tian & Yanping Chen & Jianyun Wang, 2023. "Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation," Mathematics, MDPI, vol. 11(14), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3127-:d:1194618
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