Real‐Space and Multigrid Methods in Computational Chemistry

Beck, Thomas L.

doi:10.1002/9780470399545.ch5

Cited by 8 publications

(10 citation statements)

References 241 publications

(257 reference statements)

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“…Applications of the finite element method [25] to the electronic structure of atoms and molecules go back to the 1970s [26,27] (see, e.g., Ref. [21] for a review). Applications to condensed matter systems appeared about a decade later [7,[28][29][30], and have been in active development by a number of groups since then [6,7,[29][30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

“…The advantages of a strictly local, real-space approach in large-scale calculations have been amply demonstrated in the context of finite-difference (FD) methods [5,6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23]. These methods allow for some variable resolution in real space, can accommodate a variety of boundary conditions, and require no computation-or communication-intensive transforms.…”

Section: Introductionmentioning

confidence: 99%

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Partition of unity finite element method for quantum mechanical materials calculations

Pask

Sukumar

2017

Extreme Mechanics Letters

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The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires significant nonlocal communications, which limit parallel efficiency. Real-space methods such as finite-differences (FD) and finite-elements (FE) have partially addressed both resolution and parallel-communications issues but have been plagued by one key disadvantage relative to PW: excessive number of degrees of freedom (basis functions) needed to achieve the required accuracies. In this paper, we present a real-space partition of unity finite element (PUFE) method to solve the Kohn-Sham equations of density functional theory. In the PUFE method, we build the known atomic physics into the solution process using partition-of-unity enrichment techniques in finite element analysis. The method developed herein is completely general, applicable to metals and insulators alike, and particularly efficient for deep, localized potentials, as occur in calculations at extreme conditions of pressure and temperature. Full self-consistent Kohn-Sham calculations are presented for LiH, involving light atoms, and CeAl, involving heavy atoms with large numbers of atomic-orbital enrichments. We find that the new PUFE approach attains the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the PW method. We compute the equation of state of LiH and show that the computed lattice constant and bulk modulus are in excellent agreement with reference PW results, while requiring an order of magnitude fewer degrees of freedom to obtain.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Partition of unity finite element method for quantum mechanical materials calculations

Pask

Sukumar

2017

Extreme Mechanics Letters

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“…. Different from the source problems (such as the Poisson equation for the electrostatic energy), the coarsest level must possess enough resolution to at least approximately represent orbitals . We note that the authors of ref.…”

Section: Finite Element Basis and Multigrid Preconditioningmentioning

confidence: 99%

“…Different from the source problems (such as the Poisson equation for the electrostatic energy), the coarsest level must possess enough resolution to at least approximately represent orbitals. [23] We note that the authors of ref. [36] suggested a more elaborated preconditioner of the form a À1 I þ L À1 ½M À a À1 L, where a is an estimate of the largest eigenvalue of the Hamiltonian on the coarsest mesh.…”

Section: Geometric Multigrid Preconditioner and Matrix-free Operator mentioning

confidence: 99%

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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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Overview of Electronic Structure Methods

Götz

2016

Electronic Structure Calculations on Graphics Processing Units

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Real‐Space and Multigrid Methods in Computational Chemistry

Cited by 8 publications

References 241 publications

Partition of unity finite element method for quantum mechanical materials calculations

Partition of unity finite element method for quantum mechanical materials calculations

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

Overview of Electronic Structure Methods

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