Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 12; pp. 12759 - 12776
• Main Authors: Tian, Zhikun, Chen, Yanping, Huang, Yunqing, Wang, Jianyun
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.08.2023
• Subjects: Algorithms; Elliptic functions; Error analysis; Exact solutions; Finite element method; Grid method; mixed finite element methods; Schrodinger equation; Schrödinger equation; semidiscrete scheme; Time dependence; two‐grid method
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9210

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in L2$$ {L}^2 $$‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in L2$$ {L}^2 $$‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.