@Article{NMTMA-13-2,
author = {Wang, Jianyun and Jin, Jicheng and Zhikun, Tian},
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 = {<p style="text-align: justify;">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 $H = \mathcal{O}(h^{\frac{1}{2}})$.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2019-0158},
url = {https://global-sci.com/article/90349/two-grid-finite-element-method-with-crank-nicolson-fully-discrete-scheme-for-the-time-dependent-schrodinger-equation}
}