The semiclassical Schrodinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrodinger equation is approximated by these localized multiscale basis in space and is solved by the Crank-Nicolson method in time. These multiscale basis have compact supports in space, leading to the sparsity of stiffness matrix, and thus the computational complexity of these two methods in the online stage is comparable to that of the standard finite element method. However, the spatial mesh size in multiscale finite element methods is $ H=\mathcal{O}(\epsilon) $, while $H=\mathcal{O}(\epsilon^{3/2})$ in the standard finite element method, where $\epsilon$ is the semiclassical parameter. By a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude.

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具有时间相关势的半经典薛定谔方程的有效多尺度方法

具有时间相关电位的半经典薛定谔方程是研究平均场图中外部控制下的电子动力学的重要模型。在本文中，我们提出了两种多尺度有限元方法来解决这个问题。在离线阶段，对于第一种方法，在初始时间使用哈密顿算子的稀疏压缩构建局部多尺度基函数；对于后者，稍后使用贪婪算法对哈密顿算子进行稀疏压缩，进一步丰富了基函数。在在线阶段，薛定谔方程在空间上由这些局部多尺度基逼近，并用Crank-Nicolson方法及时求解。这些多尺度基在空间上具有紧密的支撑，导致刚度矩阵的稀疏性，因此这两种方法在在线阶段的计算复杂度与标准有限元方法的计算复杂度相当。然而，多尺度有限元方法中的空间网格尺寸为$H=\mathcal{O}(\epsilon) $，而标准有限元方法中的$H=\mathcal{O}(\epsilon^{3/2})$ element 方法，其中 $\epsilon$ 是半经典参数。通过一维和二维的大量数值例子，对于近似相同数量的基，我们表明多尺度有限元方法的近似误差至少比标准有限元方法小两个数量级，并且浓缩进一步将误差降低了一个数量级。多尺度有限元方法中的空间网格尺寸为$H=\mathcal{O}(\epsilon) $，而标准有限元方法中的$H=\mathcal{O}(\epsilon^{3/2})$ ，其中 $\epsilon$ 是半经典参数。通过一维和二维的大量数值例子，对于近似相同数量的基，我们表明多尺度有限元方法的近似误差至少比标准有限元方法小两个数量级，并且浓缩进一步将误差降低了一个数量级。多尺度有限元方法中的空间网格尺寸为$H=\mathcal{O}(\epsilon) $，而标准有限元方法中的$H=\mathcal{O}(\epsilon^{3/2})$ ，其中 $\epsilon$ 是半经典参数。通过一维和二维的大量数值例子，对于近似相同数量的基，我们表明多尺度有限元方法的近似误差至少比标准有限元方法小两个数量级，并且浓缩进一步将误差降低了一个数量级。