Parallel finite element density functional computations exploiting grid refinement and subspace recycling

Young, Toby D.; Romero, Eloy; Román, José E.

doi:10.1016/j.cpc.2012.08.011

Cited by 4 publications

(13 citation statements)

References 20 publications

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“…A finite element (FE) basis set has been used in the context of the Kohn–Sham (KS) formulation of DFT in a number of works . The FE method represents a complete basis set composed of interpolation polynomials, which are related to a discrete mesh representing real space.…”

Section: Introductionmentioning

confidence: 99%

“…Let us name some of the advanced techniques, which have been used in the context of the FE solution to the KS equations. The ability to construct a hierarchy of FE spaces has been successfully utilised to speed‐up computations . The usage of spectral‐element basis functions together with the Gauss–Lobatto–Legendre quadrature rule lead to a diagonal mass matrix, thus allowing to use a Chebyshev filtering approach to compute the occupied eigenspace.…”

Section: Introductionmentioning

confidence: 99%

“…On the one hand, an interest in FE solution of the KS equations has raised a number of significant scientific publications . On the other hand, this has led to numerous different strategies considered by authors during implementation of the FE solution.…”

Section: Introductionmentioning

confidence: 99%

“…Table presents a brief overview of a chosen set of these works. Perhaps the most crucial difference between these authors' strategies is related to the FE‐trial space chosen, which ranges from the standard Lagrange polynomials of different degrees to spectral FEs .…”

Section: Introductionmentioning

confidence: 99%

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On the adaptive finite element analysis of the Kohn–Sham equations: methods, algorithms, and implementation

Davydov

Young

Steinmann

2015

Numerical Meth Engineering

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SUMMARYIn this paper, details of an implementation of a numerical code for computing the Kohn-Sham equations are presented and discussed. A fully self-consistent method of solving the quantum many-body problem within the context of density functional theory using a real-space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at coaligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon-monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the adaptive finite element analysis of the Kohn–Sham equations: methods, algorithms, and implementation

Davydov

Young

Steinmann

2015

Numerical Meth Engineering

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“…The work [Young et al 2013] presents a comparison of different refinement strategies for the meshes generated during the discretization, with a homemade code based on deal.II and SLEPc. The computation of the smallest eigenvalues (corresponding to the lowest energy orbitals) of the generalized Hermitian problems was done using GD with harmonic extraction and block Jacobi as the preconditioner for the expansion.…”

Section: Application 3: Electronic Configuration Of Atomsmentioning

confidence: 99%

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

Romero

Román

2014

ACM Trans. Math. Softw.

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In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidsontype methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

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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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Parallel finite element density functional computations exploiting grid refinement and subspace recycling

Cited by 4 publications

References 20 publications

On the adaptive finite element analysis of the Kohn–Sham equations: methods, algorithms, and implementation

On the adaptive finite element analysis of the Kohn–Sham equations: methods, algorithms, and implementation

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

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

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