Adaptive Finite Element Approximations for Kohn-Sham Models

TL;DR

This paper develops and analyzes an adaptive finite element method for solving the Kohn-Sham equations, providing convergence proofs, optimal complexity, and demonstrating robustness through numerical experiments in electronic structure calculations.

Contribution

It introduces a residual-based a posteriori error estimator and an adaptive algorithm with proven convergence and optimal complexity for Kohn-Sham models.

Findings

Convergence of the adaptive finite element method is proven.

The method achieves quasi-optimal complexity.

Numerical experiments confirm robustness and efficiency.

Abstract

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using D{\" o}rfler's marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory,but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.