Higher-order adaptive finite-element methods for Kohn-Sham density functional theory

TL;DR

This paper introduces an adaptive higher-order finite-element method for Kohn-Sham DFT calculations, achieving significant computational efficiency improvements over traditional methods, enabling large-scale electronic structure simulations.

Contribution

The paper develops an a-priori mesh adaption technique and an efficient spectral finite-element solution strategy, significantly enhancing computational efficiency for Kohn-Sham DFT calculations.

Findings

Achieves 100-200 fold speedup over generalized eigenvalue problem solutions.

Realizes up to 1000 fold computational savings compared to linear finite-elements.

Successfully computes electronic structure of a 1688-atom metallic system.

Abstract

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a-priori mesh adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100-200 fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretization of the Kohn-Sham DFT problem. Our studies suggest that staggering computational savings of the order of 1000 fold relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.