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Based on the numerical method proposed in [G. Hu, X. Xie, F. Xu, J. Comput.Phys., 355 (2018), 436-449.] for Kohn-Sham equation, further improvement on theefficiency is obtained in this paper by i). designing a numerical method withthe strategy of separately handling the nonlinear Hartree potential andexchange-correlation potential, and ii).parallelizing the algorithm in aneigenpairwise approach. The feasibility of two approaches are analyzed indetail, and the new algorithm is described completely. Compared with previousresults, a significant improvement of numerical efficiency can be observed fromplenty of numerical experiments, which make the new method more suitable forthe practical problems.

On accelerating a multilevel correction adaptive finite element method for Kohn-Sham equation