摘要
针对一类二维依赖于时间的线性薛定谔方程,在空间方向采用双线性有限元进行离散,时间方向利用向后欧拉方法得到全离散有限元格式,构造一种全离散有限元两层网格算法,对薛定谔方程耦合的实部和虚部进行解耦。从而将在细网格上进行求解,简化为在粗网格上求解原问题以及在细网格上求解两个泊松方程。数值实验结果表明,两层网格有限元方法比标准有限元方法更高效,且当粗细网格尺寸满足一定条件时,数值解具有相同的最优误差阶。
In view of a class of two-dimensional time-dependent linear Schrodinger equation,the bilinear finite element method is used for a discretization in space in conjunction with backward Euler method used to obtain the full discrete finite element scheme in the temporal direction,thus constructing a two-grid algorithm for fully discrete finite element to decouple the real and imaginary parts of the coupled Schrodinger equation.Thus,the solution on fine grids is simplified as solving the original problem on coarse grids and solving two Poisson equations on fine grids.The numerical results show that the two-grid algorithm for fully discrete finite element is more efficient than the standard finite element method,and the numerical solution has the same optimal error order with the size of coarse and fine meshes meeting certain conditions.
作者
王建云
田智鲲
张丹
WANG Jianyun;TIAN Zhikun;ZHANG Dan(College of Science,Hunan University of Technology,Zhuzhou Hunan 412007,China;College of Science,Hunan Institute of Engineering,Xiangtan Hunan 411104,China)
出处
《湖南工业大学学报》
2020年第1期19-23,共5页
Journal of Hunan University of Technology
基金
湖南省教育厅科学研究基金资助项目(18C0518)
国家自然科学基金资助项目(11747095)
湖南省自然科学基金资助项目(2017JJ3064)
湖南工业大学教育教学改革研究基金资助重点项目(2018B04)
关键词
薛定谔方程
两层网格方法
有限元方法
向后欧拉方法
Schrodinger equation
two-grid method
finite element method
backward Euler method