In this paper, a novel $h$-adaptive isogeometric solver utilizing high-order hierarchical splines is proposed to solve the all-electron Kohn--Sham equation. In virtue of the smooth nature of Kohn--Sham wavefunctions across the domain, except at the nuclear positions, high-order globally regular basis functions such as B-splines are well suited for achieving high accuracy. To further handle the singularities in the external potential at the nuclear positions, an $h$-adaptive framework based on the hierarchical splines is presented with a specially designed residual-type error indicator, allowing for different resolutions on the domain. The generalized eigenvalue problem raising from the discretized Kohn--Sham equation is effectively solved by the locally optimal block preconditioned conjugate gradient (LOBPCG) method with an elliptic preconditioner, and it is found that the eigensolver's convergence is independent of the spline basis order. A series of numerical experiments confirm the effectiveness of the $h$-adaptive framework, with a notable experiment that the numerical accuracy $10^{-3} \mathrm{~Hartree/particle}$ in the all-electron simulation of a methane molecule is achieved using only $6355$ degrees of freedom, demonstrating the competitiveness of our solver for the all-electron Kohn--Sham equation.

翻译：本文提出了一种利用高阶层次样条的新型h自适应等几何求解器，用于求解全电子Kohn-Sham方程。由于Kohn-Sham波函数在除原子核位置外的整个计算域内具有光滑特性，高阶全局正则基函数（如B样条）非常适合实现高精度计算。为处理原子核位置处外势场的奇异性，本文提出基于层次样条的h自适应框架，并设计了专用的残差型误差指示器，从而允许在计算域内采用不同分辨率。通过采用椭圆预条件子的局部最优块预条件共轭梯度法（LOBPCG）有效求解离散化Kohn-Sham方程产生的广义特征值问题，研究发现该特征值求解器的收敛性与样条基函数阶数无关。一系列数值实验验证了h自适应框架的有效性，其中甲烷分子全电子模拟实验仅使用6355个自由度即达到10^{-3} Hartree/粒子的数值精度，这充分证明了本求解器在处理全电子Kohn-Sham方程方面的竞争力。