薛定谔方程向后欧拉全离散两网格混合有限元方法

Two-Grid Methods of Backward Euler Fully Discrete Mixed Finite Element for Schrodinger Equations

针对二维依赖于时间的线性薛定谔方程,在空间方向采用混合有限元方法,时间方向利用向后欧拉方法,得到一种全离散混合有限元格式.为了将薛定谔方程耦合的实部和虚部解耦,提出了一种全离散混合有限元的两网格算法,将方程在细网格上的求解问题,简化为在一个相对更粗的网格上求解原问题以及在细网格上求解两个泊松方程,从而减小计算工作量,节省计算时间.数值实验结果验证了两网格混合有限元方法的高效性.

For the two-dimensional time-dependent linear Schrodinger equation,a fully discrete mixed finite element scheme is obtained by mixed finite element method in space in conjunction with backward Euler method in time.A two-grid algorithm of fully discrete mixed finite element is proposed to decouple the real part and imaginary part of the Schrodinger equation.With this method,the problem of solving the Schrodinger equation on the fine grid is reduced to solving the original problem on a relatively coarse grid and solving two Poisson equations on the fine grid so as to reduce the calculation workload and save the calculation time.The numerical experiment results verify the efficiency of the two-grid mixed finite element method.