Adaptive Finite Element Approximations for Kohn--Sham Models

The Kohn--Sham model is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanoscience. In this paper, we study 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 a Dörfler marking strategy, we then get the convergence rate and quasi-optimal complexity. Moreover, we demonstrate 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.(An erratum is attached.)