  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 a priori mesh adaption technique,
based on the semi-discrete (discrete in space but continuous in time) error
estimate on the time-dependent Kohn-Sham orbitals, to construct a close to
optimal finite-element discretization. Subsequently, we obtain the
full-discrete error estimate to guide our choice of the time-step. We employ
spectral finite-elements along with Gauss-Legendre-Lobatto quadrature to render
the overlap matrix diagonal, simplifying the inversion of the overlap matrix
that features in the evaluation of the discrete time-evolution operator. We use
the second-order Magnus operator as the time-evolution operator in all our
calculations. Furthermore, the action of the discrete Magnus operator,
expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtained
efficiently through an adaptive Lanczos iteration. We observe close to optimal
rates of convergence of the dipole moment with respect to spatial and temporal
discretization, for both pseudopotential and all-electron calculations. We
demonstrate a staggering 100-fold reduction in the computational time afforded
by higher-order finite-elements over linear finite-elements. We present
comparative studies, in terms of accuracy and efficiency, of our approach
against finite-difference based discretization for pseudopotential
calculations, and demonstrate significant computational savings when compared
to the finite-difference method. We also demonstrate the competence of
higher-order finite-elements for all-electron benchmark systems. Lastly, we
observe good parallel scalability of the proposed method on many hundreds of
processors.
