An h-adaptive finite element solver for the calculations of the electronic structures

Bao, Gang; Hu, Guanghui; Liu, Di

doi:10.1016/j.jcp.2012.04.002

Cited by 69 publications

(92 citation statements)

References 31 publications

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“…The approximation results imply the exponential convergence rate exp(−b 3 √ N) for second order, elliptic PDEs in polygons D ⊂ R 2 (where S denotes the set of corners of D) which are considered, for example, in [1,7,12]. Theorem 1 also implies the exponential convergence rate exp(−b 4 √ N) for hp-approximations of electron densities in DFT, due to the quasioptimality of Galerkin approximations shown, for example, in [2,4] and the references there. In this application, C denotes the set of nuclei, whose centers c ∈ S are assumed known.…”

Section: Discussionmentioning

confidence: 79%

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Exponential Convergence of Simplicial h p-FEM for H 1-Functions with Isotropic Singularities

Schwab

2015

Lecture Notes in Computational Science and Engineering

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, which are analytic in Ω \S with point singularities concentrated at the set S ⊂ Ω consisting of a finite number of points in Ω , we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on families of regular, simplicial meshes in Ω . The simplicial meshes are assumed to be geometrically refined towards S and to be shape regular, but are otherwise unstructured.

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Section: Discussionmentioning

confidence: 79%

“…We mention only nonlinear Schrödinger equations with selffocusing, density functional models in electron structure calculations (eg. [8,2,4] and the references there), nonlinear parabolic PDEs with critical growth (eg. [15] and the references there, or continuum models of crystalline solids with isolated point defects (eg.…”

Section: Introductionmentioning

confidence: 99%

Exponential Convergence of Simplicial h p-FEM for H 1-Functions with Isotropic Singularities

Schwab

2015

Lecture Notes in Computational Science and Engineering

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Kanungo

Gavini

2017

Phys. Rev. B

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We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The compact support for the enrichment functions is obtained by using smooth cutoff functions, which enhances the conditioning and maintains the locality of the enriched finite element basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid that is constructed based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules, which is required to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem, in each self-consistent field iteration, by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavymetallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are ∼ 1 mHa with reference ground-state energies employing classical finite element as well as gaussian basis sets. Using the proposed formulation based on enriched finite element basis, for accuracies commensurate with chemical accuracy, we observe a staggering 50−300 fold reduction in the overall computational time when compared to classical finite element basis. Further, we find a significant outperformance by the enriched finite element basis when compared to the gaussian basis for the modest system sizes where we obtained convergence with gaussian basis. We also observe good parallel scalability of the numerical implementation up to 384 processors for a representative benchmark system comprising of 280-atom silicon nano-cluster.

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“…Further, the finite-element basis is amenable to adaptive spatial resolution which can effectively be exploited for efficient solution of all-electron DFT calculations [23][24][25] as well as the development of coarse-graining techniques that seamlessly bridge electronic structure calculations with continuum 26,27 . There have been significant efforts in the recent past to develop real-space electronic structure calculations based on a finite-element discretization 14,[23][24][25][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] .…”

Section: Introductionmentioning

confidence: 99%

Subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization

Motamarri

Gavini

2014

Phys. Rev. B

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We present a subspace projection technique to conduct large-scale Kohn-Sham density functional theory calculations using higher-order spectral finite-element discretization. The proposed method treats both metallic and insulating materials in a single framework, and is applicable to both pseudopotential as well as all-electron calculations. The key ideas involved in the development of this method include: (i) employing a higher-order spectral finite-element basis that is amenable to mesh adaption; (ii) using a Chebyshev filter to construct a subspace which is an approximation to the occupied eigenspace in a given self-consistent field iteration; (iii) using a localization procedure to construct a non-orthogonal localized basis spanning the Chebyshev filtered subspace; (iv) using a Fermi-operator expansion in terms of the subspace-projected Hamiltonian represented in the non-orthogonal localized basis to compute relevant quantities like the density matrix, electron density and band energy. We demonstrate the accuracy and efficiency of the proposed approach on benchmark systems involving pseudopotential calculations on aluminum nano-clusters up to 3430 atoms and on alkane chains up to 7052 atoms, as well as all-electron calculations on silicon nanoclusters up to 3920 electrons. The benchmark studies revealed that accuracies commensurate with chemical accuracy can be obtained with the proposed method, and a subquadratic-scaling with system size was observed for the range of materials systems studied. In particular, for the alkane chains-representing an insulating material-close to linear-scaling is observed, whereas, for aluminum nano-clusters-representing a metallic material-the scaling is observed to be O(N 1.46 ). For all-electron calculations on silicon nano-clusters, the scaling with the number of electrons is computed to be O(N 1.75 ). In all the benchmark systems, significant computational savings have been realized with the proposed approach, with ∼ 10−fold speedups observed for the largest systems with respect to reference calculations.

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An h-adaptive finite element solver for the calculations of the electronic structures

Cited by 69 publications

References 31 publications

Exponential Convergence of Simplicial h p-FEM for H 1-Functions with Isotropic Singularities

Exponential Convergence of Simplicial h p-FEM for H 1-Functions with Isotropic Singularities

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

Subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization

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