In this paper, we introduce a highly accurate and efficient numerical solver for the radial Kohn--Sham equation. The equation is discretized using a high-order finite element method, with its performance further improved by incorporating a parameter-free moving mesh technique. This approach greatly reduces the number of elements required to achieve the desired precision. In practice, the mesh redistribution involves no more than three steps, ensuring the algorithm remains computationally efficient. Remarkably, with a maximum of $13$ elements, we successfully reproduce the NIST database results for elements with atomic numbers ranging from $1$ to $92$.

翻译：本文提出了一种高精度、高效率的径向Kohn-Sham方程数值求解器。该方程采用高阶有限元方法进行离散化，并通过引入无参数移动网格技术进一步提升了计算性能。该方法显著减少了达到目标精度所需的网格单元数量。在实际计算中，网格重分布过程不超过三步，确保了算法的计算效率。值得注意的是，仅使用最多$13$个网格单元，我们成功复现了原子序数$1$至$92$元素在NIST数据库中的计算结果。