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In this article, we provide a priori estimates for a perturbation-based post-processing method of the plane-wave approximation of nonlinear Kohn–Sham local density approximation (LDA) models with pseudopotentials, relying on Cancès et al. (2020, Post-processing of the plane-wave approximation of Schrödinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, draa044) for the proofs of such estimates in the case of linear Schrödinger equations. As in Cancès et al. (2016, A perturbation-method-based post-processing for the plane-wave discretization of Kohn–Sham models. J. Comput. Phys. , 307 , 446–459), where these a priori results were announced and tested numerically, we use a periodic setting and the problem is discretized with plane waves (Fourier series). This post-processing method consists of performing a full computation in a coarse plane-wave basis and then to compute corrections based on the first-order perturbation theory in a fine basis, which numerically only requires the computation of the residuals of the ground-state orbitals in the fine basis. We show that this procedure asymptotically improves the accuracy of two quantities of interest: the ground-state density matrix, i.e. the orthogonal projector on the lowest |$N$| eigenvectors, and the ground-state energy. [ABSTRACT FROM AUTHOR]

In this article we prove a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works (Cancès, Dusson, Maday, Stamm and Vohralík, (2014), A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math. , 352 , 941--946; Cancès, Dusson, Maday, Stamm and Vohralík, (2016), A perturbation-method-based postprocessing for the planewave discretization of Kohn–Sham models. J. Comput. Phys. , 307 , 446--459.) We consider here a Schrödinger operator |${{\mathscr{H}} \,}= -\frac{1}{2}\varDelta +{\mathscr{V}}$| on |$L^2(\varOmega)$|⁠ , where |$\varOmega $| is a cubic box with periodic boundary conditions and where |${\mathscr{V}}$| is a multiplicative operator by a regular-enough function |${\mathscr{V}}$|⁠. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest |$N$| eigenvalues of |${{\mathscr{H}} \,}$|⁠ , and, on the other hand, the ground-state density matrix that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn–Sham density functional theory. Interpreting the exact eigenpairs of |${{\mathscr{H}} \,}$| as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e. Fourier) basis we compute first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This allows us to increase the accuracy by a factor proportional to the inverse of the kinetic energy cutoff |${E_{\textrm{c}}}^{-1}$| of both the ground-state energy and the ground-state density matrix in Hilbert–Schmidt norm at a low computational extra cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger plane-wave basis and two fast Fourier transforms per eigenvalue. [ABSTRACT FROM AUTHOR]

The article discusses the study of a posteriori error analysis of Gross-Pitaevskii type focusing on its variational approximations of the ground state eigenvector in a nonlinear elliptic problem. The study uses Fourier spectral approximation to solve nonlinear problem wherein a Gross-Pitaevskii type problem was solved through a posteriori estimates and a priori analysis of the convergence rates.