Higher-order adaptive finite-element methods for Kohn–Sham density functional theory

Abstract

state energy, subject to a fixed number of elements in the finite-element mesh. To this end, we first develop an estimate for the finite-element discretization error in the Kohn-Sham ground-state energy as a function of the characteristic mesh-size distribution, h(r), and the exact ground-state electronic fields comprising of wavefunctions and electrostatic potential. We subsequently determine the optimal mesh distribution for the chosen representative solution by determining the h(r) that minimizes the discretization error. The resulting expressions for the optimal mesh distribution are in terms of the degree of the interpolating polynomial and the exact solution fields of the Kohn-Sham DFT problem. Since the exact solution fields are a priori unknown, we use the asymptotic behavior of the atomic wavefunctions [38] away from the nuclei to determine the coarse-graining rates for the finite-element meshes used in our numerical study. Though the resulting finite-element meshes are not necessarily optimal near the vicinity of the nuclei, the mesh coarsening rate away from the nuclei provides an efficient way of resolving the vacuum in non-periodic calculations.We next implement an efficient solution strategy for solving the finite-element discretized eigenvalue problem, which is crucial before assessing the computational efficiency of the basis. We note that the non-orthogonality of the finite-element basis results in a discrete generalized eigenvalue problem, which is computationally more expensive than the standard eigenvalue problem that results from using an orthogonal basis like planewaves. We address this issue by employing a spectral finite-element discretization and Gauss-Lobatto quadrature rules to evaluate the integrals which results in a diagonal overlap matrix, and allows for a trivial transformation to a standard eigenvalue problem. Further, we use the Chebyshev acceleration technique for standard eigenvalue problems to efficiently compute the occupied eigenspace (cf. e.g. [39] in the context of electronic structure calculations). Our investigations suggest that the use of spectral finite-elements and Gauss-Lobatto rules in conjunction with Chebyshev acceleration techniques to compute the eigenspace gives a 10 − 20 fold computational advantage, even for modest materials system sizes, in comparison to traditional methods of solving the standard eigenvalue problem where the eigenvectors are computed explicitly. Further, the proposed approach has been observed to provide a staggering 100 − 200 fold computational advantage over the solution of a generalized eigenvalue problem that does not take advantage of the spectral finite-element discretization and Gauss-Lobatto quadrature rules. In our implementation, we use a self-consistent field (SCF) iteration with Anderson mixing [40], and employ the finite-temperature Fermi-Dirac smearing [3] to suppress the charge sloshing associated with degenerate or close to degenerate eigenstates around the Fermi energy.We next study various numerical aspects of the finite-...

Electronic Structure Calculations at Macroscopic Scales

“…We conducted benchmark studies to compare the performance of our implementation with existing softwares like ABINIT and Gaussian. The proposed approach to the solution of the Kohn-Sham DFT problem with higher-order FE basis functions is comparing favorably with existing highly optimized commercial codes (Motamarri et al, 2012). Moreover, we note that the finite-element discretization offers many other advantages, which include the ability to handle complex geometries and boundary conditions, good scalability on parallel computing architectures, and the ability to handle both pseudopotential and all-electron calculations.…”

“…In particular, we developed a framework for higher order finite-element discretizations which include quadratic tetrahedral FE basis functions, and hexahedral node-based Lagrange basis functions up to 10th order which includes spectral elements. We conducted systematic studies to assess the computational efficiency afforded by higher-order FE discretizations for both orbital-free DFT and Kohn-Sham DFT (Motamarri et al, 2012). Our benchmark numerical studies demonstrated optimal rates of convergence for these discretization based on linear finite-element analysis although the present problem is non-linear.…”

“…The techniques have been developed for orbital-free DFT (Wang & Teter, 1992;Wang et al, 1998Wang et al, , 1999, which is an approximation to the Kohn-Sham density functional theory where the kinetic energy functionals are modeled and is applicable to materials systems whose electronic structure is close to a free electron gas. Further, in order to extend these techniques to Kohn-Sham density functional theory, we have developed a real-space formulation of Kohn-Sham density functional theory (Suryanarayana et al, 2010;Motamarri et al, 2012), which is the basis for the coarse-graining techniques based on quasi-continuum reduction (Gavini et al, 2007;1 Development of quasi-continuum reduction of orbitalfree DFT with non-local kinetic energy functionals and Mathematical Analysis:…”

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

“…Finite element basis 42,43 , on the other hand, being a local piecewise polynomial basis, retains the variational property of the plane-waves, and, in addition, has other desirable features such as locality of the basis that affords good parallel scalability, being easily amenable to adaptive spatial resolution, and the ease of handling arbitrary boundary conditions. While most studies employing the finite element basis in DFT calculations [44][45][46][47][48][49][50][51][52][53] have shown its usefulness in pseudopotential calculations, some of the works 44,[53][54][55][56][57] have also demonstrated its promise for all-electron calculations. In particular, the work of Motamarri et.…”

“…In particular, the work of Motamarri et. al 53 has combined the use of higher-order spectral finite elements along with Chebyshev polynomial based filtering technique to develop an efficient scheme for the computation of the occupied eigenstates. As detailed in the work, the aforementioned method outperforms the plane-wave basis in pseudopotential calculations for the benchmark systems considered.…”