In this Note, we propose a new method, based on perturbation theory, to post-process the planewave approximation of the eigenmodes of periodic Schrödinger operators. We then use this post-processing to construct an accurate a posteriori estimator for the approximations of the (nonlinear) Gross–Pitaevskii equation, valid at each step of a self-consistent procedure. This allows us to design an adaptive algorithm for solving the Gross–Pitaevskii equation, which automatically refines the discretization along the convergence of the iterative process, by means of adaptive stopping criteria.
Dans cette Note, nous proposons une nouvelle méthode, basée sur la théorie des perturbations, pour post-traiter l'approximation dans une base d'ondes planes des modes propres d'opérateurs de Schrödinger périodiques. Nous utilisons ensuite ce post-traitement pour construire un estimateur d'erreur a posteriori pour les approximations de l'équation de Gross–Pitaevskii (non linéaire), valide à chaque étape de la procédure auto-cohérente. Ceci nous permet de proposer un algorithme adaptatif pour résoudre cette équation, qui raffine automatiquement la discrétisation au cours du processus itératif, par le biais de critères d'arrêt adaptatifs.
Accepted:
Published online:
Éric Cancès 1; Geneviève Dusson 2, 3; Yvon Maday 2, 4, 5; Benjamin Stamm 2; Martin Vohralík 6
@article{CRMATH_2014__352_11_941_0,
author = {\'Eric Canc\`es and Genevi\`eve Dusson and Yvon Maday and Benjamin Stamm and Martin Vohral{\'\i}k},
title = {A perturbation-method-based \protect\emph{a posteriori} estimator for the planewave discretization of nonlinear {Schr\"odinger} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {941--946},
publisher = {Elsevier},
volume = {352},
number = {11},
year = {2014},
doi = {10.1016/j.crma.2014.09.014},
language = {en},
}
TY - JOUR AU - Éric Cancès AU - Geneviève Dusson AU - Yvon Maday AU - Benjamin Stamm AU - Martin Vohralík TI - A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations JO - Comptes Rendus. Mathématique PY - 2014 SP - 941 EP - 946 VL - 352 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2014.09.014 LA - en ID - CRMATH_2014__352_11_941_0 ER -
%0 Journal Article %A Éric Cancès %A Geneviève Dusson %A Yvon Maday %A Benjamin Stamm %A Martin Vohralík %T A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations %J Comptes Rendus. Mathématique %D 2014 %P 941-946 %V 352 %N 11 %I Elsevier %R 10.1016/j.crma.2014.09.014 %G en %F CRMATH_2014__352_11_941_0
Éric Cancès; Geneviève Dusson; Yvon Maday; Benjamin Stamm; Martin Vohralík. A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 941-946. doi : 10.1016/j.crma.2014.09.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.014/
[1] Adaptive error control for multigrid finite element methods, Computing, Volume 55 (1995), pp. 271-288
[2] Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput., Volume 45 (2010), pp. 90-117
[3] E. Cancès, G. Dusson, Y. Maday, B. Stamm, M. Vohralík, On the use of the perturbation method in planewave numerical simulations of linear and nonlinear Schrödinger equations, in preparation.
[4] A posteriori estimation of the linearization error for strongly monotone nonlinear operators, J. Comput. Appl. Math., Volume 205 (2007), pp. 72-87
[5] Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., Volume 35 (2013), p. A1761-A1791
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