Numerical analysis of finite dimensional approximations of Kohn–Sham models

Chen, Huajie; Gong, X. G.; He, Limin; Zhang, Yang; Zhou, Aihui

doi:10.1007/s10444-011-9235-y

Cited by 41 publications

(36 citation statements)

References 34 publications

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“…The idea here is to use the multilevel correction method to transform the solution of the nonlinear eigenvalue problem to a series of solutions of the corresponding linear boundary value problems with multigrid method and a series of nonlinear eigenvalue problems on the coarsest finite element space. The proposed multigrid method can be applied to practical nonlinear eigenvalue problems [6,7,8,9]. We can replace the multigrid method by other types of efficient iteration schemes such as algebraic multigrid method, the type of preconditioned schemes based on the subspace decomposition and subspace corrections (see, e.g., [5,25]), and the domain decomposition method (see, e.g., [21]).…”

Section: Discussionmentioning

confidence: 99%

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A multigrid method for nonlinear eigenvalue problems

Xie¹

2015

Sci. Sin.-Math.

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A multigrid method is proposed for solving nonlinear eigenvalue problems by the finite element method. With this new scheme, solving nonlinear eigenvalue problem is decomposed to a series of solutions of linear boundary value problems on multilevel finite element spaces and a series of small scale nonlinear eigenvalue problems. The computational work of this new scheme can reach almost the same as the solution of the corresponding linear boundary value problem. Therefore, this type of multilevel correction scheme improves the overfull efficiency of the nonlinear eigenvalue problem solving.

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Section: Discussionmentioning

confidence: 99%

“…For more discussions about the function f (x, ·), please refer to [6,7,27] and the papers cited therein.…”

Section: Assumption Bmentioning

confidence: 99%

A multigrid method for nonlinear eigenvalue problems

Xie¹

2015

Sci. Sin.-Math.

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“…Note that we have not obtained a priori error estimates for approximations of nonlinear eigenvalue problems but only for linearized equations in SCF iterations. We refer to [13,17] for numerical analysis of nonlinear eigenvalue problems.…”

Section: Dg Approximations Of the Eigenvalue Problemmentioning

confidence: 99%

A discontinuous Galerkin scheme for full-potential electronic structure calculations

Chen

2019

Journal of Computational Physics

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In this paper, we construct an efficient numerical scheme for full-potential electronic structure calculations of periodic systems. In this scheme, the computational domain is decomposed into a set of atomic spheres and an interstitial region, and different basis functions are used in different regions: radial basis functions times spherical harmonics in the atomic spheres and plane waves in the interstitial region. These parts are then patched together by discontinuous Galerkin (DG) method. Our scheme has the same philosophy as the widely used (L)APW methods in materials science, but possesses systematically spectral convergence rate. We provide a rigorous a priori error analysis of the DG approximations for the linear eigenvalue problems, and present some numerical simulations in electronic structure calculations.

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“…The efficacy of the finite-element basis in terms of its accuracy, efficiency, scalability and relative performance with other competing methods (e.g., planewaves, Gaussian basis, FD), have been thoroughly studied in the context of ground-state DFT, for both pseudopotential [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] and all-electron calculations 37,48,[50][51][52][53][54][55][56][57][58] . A similarly comprehensive study on the efficacy of the finite-element basis for RT-TDDFT is, however, lacking.…”

Section: Introductionmentioning

confidence: 99%

Real time time-dependent density functional theory using higher order finite-element methods

Kanungo

Gavini

2019

Phys. Rev. B

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We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real-time using higher-order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an a priori mesh adaption technique, based on the semi-discrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time-step. We employ spectral finite-elements along with special reduced order quadrature to render the overlap matrix diagonal, thereby simplifying the inversion of the overlap matrix that features in the evaluation of the discrete time-evolution operator. We use the second-order Magnus operator as the time-evolution operator, wherein the action of the discrete Magnus operator, expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtained efficiently through an adaptive Lanczos iteration. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretization, for both pseudopotential and all-electron calculations. We demonstrate a staggering 100-fold reduction in the computational time afforded by higher-order finite-elements over linear finite-elements, for both pseudopotential and all-electron calculations. Further, for similar level of accuracy, we obtain significant computational savings by our approach as compared to state-of-theart finite-difference methods. We also demonstrate the competence of higher-order finite-elements for all-electron benchmark systems. Lastly, we observe good parallel scalability of the proposed method on many hundreds of processors.

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Numerical analysis of finite dimensional approximations of Kohn–Sham models

Cited by 41 publications

References 34 publications

A multigrid method for nonlinear eigenvalue problems

A multigrid method for nonlinear eigenvalue problems

A discontinuous Galerkin scheme for full-potential electronic structure calculations

Real time time-dependent density functional theory using higher order finite-element methods

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