第一原理计算研究：双尺度离散和预处理迭代法

The first-principles calculations enable one to study electronic structures of materials. One can simulate and even predict structure and nature of materials by the first-principles calculations. We will study the high efficient numerical methods for solving the Kohn-Sham equations and the Sternheimer equations which are very important in the first-principles calculations. Firstly, the ground state properties of materials can be described by the Kohn-Sham equations very well. The Kohn-Sham equations are nonlinear eigenvalue problems. We will design and analyze postprocessed two-scale finite element discretizations for the Kohn-Sham equations and the related nonlinear boundary value problems. Secondly, the excited state properties of materials can be described by the first-principles GW method properly. The Sternheimer equations, which are non-SPD linear elliptic problem, are important in the first-principles GW method. We will design a two-grid method for the Sternheimer equations based on the plane-wave discretization in the software BerkeleyGW, in which the first-principles GW method is implemented. Because the linear systems from the Sternheimer equations at some frequencies can be ill-conditioned, we will develop proper preconditioned iterative method to improve the computational efficiency.

第一原理计算使人们能深入研究物质的电子结构，模拟甚至预测材料的结构和性能。我们拟针对第一原理计算中的两类重要方程---Kohn-Sham方程和Sternheimer方程---研究高效数值计算方法。首先，Kohn-Sham方程能很好地描述材料的基态性质。该方程是一类非线性特征值问题。我们拟针对Kohn-Sham方程及其相关的非线性边值问题设计和分析后处理双尺度有限元离散方法。第二，第一原理GW方法能合适地描述材料的激发态性质。Sternheimer方程是第一原理GW方法中的重要方程。该方程是非对称正定的线性椭圆方程，我们拟在第一原理GW计算软件包BerkeleyGW中针对该方程设计基于平面波离散的双网格方法。在某些频率值上Sternheimer方程离散后得到的线性方程组是病态的，我们拟对该方程组设计合适的预处理迭代法以提高计算效率。

第一原理计算已在解释实验现象、预测材料性质和设计新材料等方面取得了巨大成功。在第一原理计算中设计和运用高效的数值计算方法能让人们更好地研究更大规模、更复杂的体系。本项目取得了如下进展：针对第一原理计算中的一类非线性特征值问题设计了一类双尺度有限元离散方法；针对一类半线性抛物问题设计了双尺度有限元离散方法；采用第一原理计算软件包ABINIT，研究了金原子链的稳定性；提出了交替半正定分裂预处理子来求解由电磁场时谐涡流模型离散所得到的鞍点问题；建立了一类改进的松弛半正定和反对称分裂预处理子求解鞍点问题；提出了局部正定与反对称分裂迭代法求解非对称鞍点问题，证明了该方法的收敛性。