In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.

翻译：本文提出了一种基于移动网格（非嵌套网格）自适应技术和增广子空间方法的自适应有限元新方法，用于求解Kohn-Sham方程。与需要在每个自适应有限元空间中直接求解Kohn-Sham方程的经典自洽场迭代算法不同，本算法将Kohn-Sham方程转化为每个自适应有限元空间中规模相同的线性边值问题，然后通过在低维增广子空间中求解一个小规模Kohn-Sham方程，对线性边值问题得到的波函数进行修正。由于新算法避免了直接求解大规模Kohn-Sham方程，因此求解效率得到显著提升。此外，根据近似波函数的奇异性，采用自适应移动网格技术为非嵌套增广子空间方法生成非嵌套自适应网格。利用近似波函数的修正Hessian矩阵作为度量矩阵重新分布网格。通过移动网格自适应技术，重新分布的网格几乎达到最优。通过大量数值实验验证了所提算法的效率和精度。