A Nonnested Augmented Subspace Method for Kohn-Sham Equation

TL;DR

This paper introduces a novel adaptive finite element method combining moving mesh and augmented subspace techniques to efficiently solve the Kohn-Sham equation, significantly reducing computational costs while maintaining accuracy.

Contribution

The paper presents a new nonnested augmented subspace method with moving mesh adaptation for solving the Kohn-Sham equation more efficiently than classical approaches.

Findings

Significant reduction in computational time.

High accuracy in wavefunction approximation.

Effective mesh redistribution using modified Hessian matrix.

Abstract

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.