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

Zhikun Tian, Yanping Chen and Jianyun Wang ()
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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

Mathematics, 2023, vol. 11, issue 14, 1-14

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.

Keywords: two-grid method; Schrödinger equation; mixed finite element method; backward Euler scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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