Finite element method for solving Kohn–Sham equations based on self-adaptive tetrahedral mesh

Zhang, Dier; Shen, Lihua; Zhou, Aihui; Gong, X. G.

doi:10.1016/j.physleta.2008.05.075

Cited by 52 publications

(34 citation statements)

References 23 publications

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“…This is in contrast to a substantial amount of global communication in the plane waves basis with the fast Fourier transform . We note that the FEM has been used as a spatial discretization technique both in the case of Self Consistent Field solution of the DFT as well as the direct minimization of the energy functional . It can also be applied to infinite systems with periodic boundary conditions …”

Section: Introductionmentioning

confidence: 99%

Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning

Davydov

Heister

Kronbichler

et al. 2018

Physica Status Solidi (b)

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In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h‐adaptive finite element method is adopted for spatial discretization and implemented with matrix‐free operator evaluation. The ground state energy is found by performing unconstrained minimization with non‐orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward O(N) complexity for the locally adaptive real‐space solution of density functional theory with finite elements.

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Section: Introductionmentioning

confidence: 99%

Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning

Davydov

Heister

Kronbichler

et al. 2018

Physica Status Solidi (b)

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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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“…People may refer to [23] for the finite element methods in the electronic structure calculations. Although the finite element methods have been developed for the groundstate calculations of the electronic system in [29,35,2,1,3,10,17], etc., little is known about the application of finite element methods for the RT-TDKS. In [27], a finite element method solver for time-dependent and stationary Schrödinger equations is proposed.…”

Section: Introductionmentioning

confidence: 99%

“…In the simulation, the mesh density is adjusted dynamically according to the numerical results with the mesh adaptive methods, and computational resource can be potentially saved. Although there have been a lot of works on developing mesh adaptive methods for ground-state calculations [10,35,8], rare of them is related to the RT-TDKS simulations.…”

Section: Introductionmentioning

confidence: 99%

Real-time adaptive finite element solution of time-dependent Kohn–Sham equation

Bao

Liu

2015

Journal of Computational Physics

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Finite element method for solving Kohn–Sham equations based on self-adaptive tetrahedral mesh

Cited by 52 publications

References 23 publications

Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning

Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning

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

Real-time adaptive finite element solution of time-dependent Kohn–Sham equation

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