二维薛定谔方程的全离散有限元两层网格方法

A Two-Grid Algorithm for Fully Discrete Finite Element Based on the Two-Dimensional Schrodinger Equation

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.