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 - <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>