Higher-Order Adaptive Finite-Element Methods for Kohn-Sham Density Functional Theory

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a priori mesh adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretizations of the Kohn-Sham DFT problem. Our studies suggest that staggering computational savings--of the order of 1000-fold--can be realized, for both all-electron and pseudopotential calculations, by using higher-order finite-element discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element discretization of the Kohn-Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic pseudopotential calculations, and is comparable to the Gaussian basis for all-electron calculations. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of materials systems containing a few thousand atoms using modest computational resources, and good scalability of the present implementation up to a few hundred processors. Preprint. Journal to be determined. Prepared in collaboration with Department of Mechanical Engineering, University of Michigan, Ann Arbor. Sponsored in part by AFOSR. The original document contains color images.