  In the present work, we investigate the computational efficiency afforded by
higher-order finite-element discretization of the saddle-point formulation of
orbital-free density functional theory. We first investigate the robustness of
viable solution schemes by analyzing the solvability conditions of the discrete
problem. We find that a staggered solution procedure where the potential fields
are computed consistently for every trial electron-density is a robust solution
procedure for higher-order finite-element discretizations. We next study the
numerical convergence rates for various orders of finite-element approximations
on benchmark problems. We obtain close to optimal convergence rates in our
studies, although orbital-free density-functional theory is nonlinear in nature
and some benchmark problems have Coulomb singular potential fields. We finally
investigate the computational efficiency of various higher-order finite-element
discretizations by measuring the CPU time for the solution of discrete
equations on benchmark problems that include large Aluminum clusters. In these
studies, we use mesh coarse-graining rates that are derived from error
estimates and an a priori knowledge of the asymptotic solution of the far-field
electronic fields. Our studies reveal a significant 100-1000 fold computational
savings afforded by the use of higher-order finite-element discretization,
alongside providing the desired chemical accuracy. We consider this study as a
step towards developing a robust and computationally efficient discretization
of electronic structure calculations using the finite-element basis.
