Development of real-space ab-initio calculation based on finite-element method aiming for nano-interface structural optimization.

为了开发一种快速的从头计算软件，研究了实际空间从头计算中的有限元技术，特别是形状函数和网格划分方法。通过研究，我们发现了以下问题：(1)弯曲网格法即使在三维情况下也保持了有限元法的收敛性，有助于减少网格点的数量。(2)与低阶形状函数相比，谱元法生成的高阶形状函数网格点数量更少，结果更准确。(3)曲线网格的四次偶然性有限元比规则网格的二次偶然性有限元能显著提高计算效率。

For the development of a fast ab initio calculation software, the finite-element techniques, especially shape functions and meshing methods, in real-space ab initio calculation have been investigated. Through the investigation, we revealed the following issues : (1) Curving-mesh method maintains the convergence property of the finite-element method even in the 3-dimensional case and proves useful to reduce the number of grid points. (2) Higher-order shape function generated by spectral element method yields sufficiently accurate results with fewer number of grid points compared to lower-order shape function. (3) Quartic serendipity finite-element with curving grid can significantly improve the computational efficiency compared to the quadratic serendipity element with regular grid.