Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation
Year: 2020
Author: Jianyun Wang, Jicheng Jin, Zhikun Tian
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 2 : pp. 334–352
Abstract
In this paper, we study the Crank-Nicolson Galerkin finite element method and construct a two-grid algorithm for the general two-dimensional time-dependent Schrödinger equation. Firstly, we analyze the superconvergence error estimate of the finite element solution in $H^1$ norm by use of the elliptic projection operator. Secondly, we propose a fully discrete two-grid finite element algorithm with Crank-Nicolson scheme in time. 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 Poisson equations on the fine grid. Finally, we also derive error estimates of the two-grid finite element solution with the exact solution in $H^1$ norm. It is shown that the solution of two-grid algorithm can achieve asymptotically optimal accuracy as long as mesh sizes satisfy $H = \mathcal{O}(h^{\frac{1}{2}})$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2019-0158
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 2 : pp. 334–352
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Schrödinger equations two-grid algorithms Crank-Nicolson scheme finite element method.