Multigrid Electrostatic Computations in Density Functional Theory

Merrick, Michael P.; Iyer, Karthik A.; Beck, Thomas L.

doi:10.1007/978-3-662-09638-3_13

Cited by 1 publication

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“…The multigrid method (Brandt, 1977(Brandt, , 1982(Brandt, , 1984(Brandt, , 1999Hackbusch, 1985) is a particularly efficient method for solving the discrete equations. Linear-scaling real-space methods have been developed for solution of the Poisson problem in DFT (White et al, 1989;Merrick et al, 1995Merrick et al, , 1996Gygi and Galli, 1995;Briggs et al, 1995;Modine et al, 1997;Goedecker and Ivanov, 1998a). These studies have illustrated the accuracies and efficiencies of the real-space approach.…”

Section: A Classical Electrostaticsmentioning

confidence: 99%

Real-space mesh techniques in density-functional theory

2000

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This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.

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Section: A Classical Electrostaticsmentioning

confidence: 99%