Fast real-time time-dependent density functional theory calculations using higher-order finite element methods

We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real time using higher order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an apriori mesh adaption technique, based on the semidiscrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time step. Importantly, for the all-electron case, we present an efficient mixed basis, termed as enriched finite element basis, that combines the efficiency of atomic-orbitals-type basis to capture the sharp variations of the electronic fields closer to the atoms along with the completeness and spatial-adaptivity of the finite element basis. We demonstrate significant savings afforded by our approach over the finite-difference and Gaus-
sian basis based approaches, for pseudopotential and all-electron calculations, respectively. Additionally, we discuss schemes to accelerate the time-evolution by constructing dynamic subspace based approaches, informed by time-dependent perturbation theory.