线性Schrdinger方程的H^1-Galerkin混合有限元方法(英文)

H^1-Galerkin Mixed Finite Element Method for the Linear Schrdinger Equation

利用H^1-Galerkin混合元方法讨论一类二阶线性Schrdinger方程.根据方程的特点将方程的实虚部进行分离,分别对实虚部方程应用H^1-Galerkin混合元方法,并且统一考虑,得到一维情况下半离散和全离散格式的最优阶误差估计,并且推广应用到二维和三维情况,而且所提出方法不须满足LBB相容性条件.与一维情况相比二维或三维流量的L^2模误差估计不是最优的,因此提出和分析了其修正格式.最后,通过数值算例验证所提出方法的可行性.

An H^1-Galerkin mixed finite element method is discussed for a class of second-order linear Schrodinger equation.We separate the real and imaginary part of the equation according to the feature of the problem.The H^1-Galerkin mixed method is studied for the real and imaginary part respectively.The optimal error estimates of the semi-discrete and fully discrete schemes are derived by considering the two part of the equation simultaneously in one space dimension.An extension to problems in two and three space variables is also considered and the proposed method dose not require the LBB consistency condition.Compared to the results proved for one space variable,the L^2 estimate of the flux is not optimal for the problem in two and three space dimensions.So Therefore,a modification of the method is proposed and analyzed.Finally,a numerical example is presented to illustrate the effectiveness of the proposed method.