Kohn-Sham方程是描述物质微观结构特别是物质电子结构的重要模型，其特征函数具有局部的奇异性（高振荡性）。对于这一问题，目前大多数计算方法的计算复杂度相对于体系粒子数都是2次甚至3次的标度，因而很难处理大的体系。本项目将充分利用非标准有限元方法的灵活性，研究将先验知识有机的嵌入到离散空间中，构造整体自由度少但精度高的新型有限元方法；研究非线性Kohn-Sham方程特征值的下界逼近，再通过重构的方法计算特征值的上界，这样得到近似特征值一个后验估计，而且可以评估近似解的可靠性（因为同时知道上界和下界）；设计能将问题的特性和先验知识嵌入到网格剖分和加密过程中的特征值问题新型自适应有限元方法；将上述离散方法有机结合，提出非线性特征值问题新型自适应非标准有限元方法并分析其收敛性和最优复杂性。总体研究目标是在非线性Kohn-Sham方程的可靠性高精度数值方法及其理论方面取得突破。

The Kohn-Sham equation is an important model that describes the substructure of matter, in particular,the electronic structure,whose eigenfunctions have some local singularity (high oscillation). For this problem,most of existing numerical methods are not able to compute the large scale system since they are quadratic or cubic scaling methods with respect to the number of particles of the system. This projection consists of three parts. In the first part of this project, the flexibility of the nonstandard finite element methods will be explored to design new schemes where the a priori knowledge will be possibly incorporated into the finite element spaces. This may result in a new method with the small degrees of freedom but high accuracy. The lower bounds of the eigenvalues for the Kohn-Sham equation will be equally investigated, this will lead to an a posteriori error estimate for the approximate eigenvalue after the upper bounds of the eigenvalues are derived based on some postprocessing. In terms of the lower and upper bounds, the reliability of the approximation solution can be evaluated. In the second part,a novel adaptive method for the eigenvalue problem will be proposed, where the a priori knowledge and the speciality of the problem are able to be incorporated into the triangulation or the refinement of the mesh. The third part will be devoted to the combination of these two new methods to construct new adaptive nonstandard finite element methods for the nonlinear eigenvalue problem. Finally, the convergence and optimality of this adaptive method will be analyzed and proved for the eigenvalue problem. The goal of this project is to make siginificant progress on the reliable high accuracy numerical methods and their corresponding theories for the nonlinear Kohn-Sham equation.