A high-order accurate moving mesh finite element method for the radial Kohn--Sham equation

TL;DR

This paper presents a high-order finite element method combined with a moving mesh technique to efficiently solve the radial Kohn--Sham equation, achieving high accuracy with minimal elements.

Contribution

The paper introduces a novel high-order finite element solver with a parameter-free moving mesh approach for the radial Kohn--Sham equation, reducing computational cost.

Findings

Achieved accurate results with only 13 elements for elements Z=1 to 92.

Reproduced NIST database results efficiently.

Mesh redistribution involves no more than three steps.

Abstract

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 $92$.