Comptes Rendus
Numerical analysis
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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.09.014
Éric Cancès 1; Geneviève Dusson 2, 3; Yvon Maday 2, 4, 5; Benjamin Stamm 2; Martin Vohralík 6

1 Université Paris-Est, CERMICS, École des ponts and INRIA, 6 & 8, av. Blaise Pascal, 77455 Marne-la-Vallée cedex 2, France
2 Sorbonne Universités, UPMC–Université Paris-6 and CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
3 Sorbonne Université, UPMC–Université Paris-6, Institut du calcul et de la simulation, 75005 Paris, France
4 Institut universitaire de France, 75005 Paris, France
5 Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA
6 INRIA Paris-Rocquencourt, domaine de Voluceau-Rocquencourt, BP 105, 78153 Le Chesnay, France
@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] R. Becker; C. Johnson; R. Rannacher Adaptive error control for multigrid finite element methods, Computing, Volume 55 (1995), pp. 271-288

[2] E. Cancès; R. Chakir; Y. Maday 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.L. Chaillou; M. Suri A posteriori estimation of the linearization error for strongly monotone nonlinear operators, J. Comput. Appl. Math., Volume 205 (2007), pp. 72-87

[5] A. Ern; M. Vohralík Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., Volume 35 (2013), p. A1761-A1791

Cited by Sources:

Comments - Policy


Articles of potential interest

KAM for quasi-linear KdV

Pietro Baldi; Massimiliano Berti; Riccardo Montalto

C. R. Math (2014)


Wavelet-based multiscale proper generalized decomposition

Angel Leon; Anais Barasinski; Emmanuelle Abisset-Chavanne; ...

C. R. Méca (2018)