Volume 13, Issue 2
Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 334-352.

Published online: 2020-03

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• 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 $H = \mathcal{O}(h^{\frac{1}{2}})$.

• Keywords

Schrödinger equations, two-grid algorithms, Crank-Nicolson scheme, finite element method.

65M15, 65M55, 65M60

wjy8137@163.com (Jianyun Wang)

jcjin2008@sina.com (Jicheng Jin)

tianzhikun@163.com (Zhikun Tian)

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@Article{NMTMA-13-334, author = {Wang , Jianyun and Jin , Jicheng and Tian , Zhikun }, title = {Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {2}, pages = {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 $H = \mathcal{O}(h^{\frac{1}{2}})$.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0158}, url = {http://global-sci.org/intro/article_detail/nmtma/15453.html} }
TY - JOUR T1 - Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation AU - Wang , Jianyun AU - Jin , Jicheng AU - Tian , Zhikun JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 334 EP - 352 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0158 UR - https://global-sci.org/intro/article_detail/nmtma/15453.html KW - Schrödinger equations, two-grid algorithms, Crank-Nicolson scheme, finite element method. AB -

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 $H = \mathcal{O}(h^{\frac{1}{2}})$.

Jianyun Wang, Jicheng Jin & Zhikun Tian. (2020). Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation. Numerical Mathematics: Theory, Methods and Applications. 13 (2). 334-352. doi:10.4208/nmtma.OA-2019-0158
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