密度泛函理论的高阶数值方法

密度泛函理论（Density Functional Theory, DFT）在电子结构计算中扮演重要角色。随着材料科学的发展和应用需求的提升，化学精度成为当前DFT计算的重要目标。然而，为达到化学精度，线性的有限元离散方法对于基函数需求非常庞大，这就要求离散方法是高阶的。而高阶离散方法可能会增加计算成本和存储需求。本项目旨在建立一个适用于DFT计算的高阶NURBS框架，并针对其中的计算困难提出相应的解决方法。一方面，引入基于调和映射的移动网格方法提升数值模拟精度，并促进NURBS框架适用DFT中更复杂的计算场景；另一方面，本项目计划针对高阶格式的计算挑战包括离散系统的构造、特征值问题的求解、线性系统的求解提出高效计算的方案。

Density functional theory (DFT) plays an important role in electronic structure calculations. With the development of materials science and the increasing demand for applications, chemical accuracy has become an important goal in current DFT calculations. However, in order to achieve chemical accuracy, linear finite element discretization methods have a huge demand for basis functions, which requires the discretization method to be higher-order. Higher-order discrete methods may increase computational costs and storage requirements. The purpose of this project is to establish a NURBS framework that is higher-order for DFT computation and to propose corresponding solutions to the computational difficulties involved. On the one hand, the moving mesh method based on the Harmonic maps is introduced to improve the accuracy of numerical simulation and promote the application of the NURBS framework to more complex computing scenarios in DFT; On the other hand, this project plans to propose efficient implementations for the computational challenges of higher-order schemes include the construction of discrete systems, the solution of eigenvalue problems, and the solution of the linear system.