A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations

Eric Cancès (1, 2) , Geneviève Dusson (3, 4) , Yvon Maday (3, 5, 6) , Benjamin Stamm (3) , Martin Vohralík (7)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 LJLL - Laboratoire Jacques-Louis Lions
4 ISCD - Institut des Sciences du Calcul et des Données
5 DAM - Division of Applied Mathematics
6 IUF - Institut universitaire de France
7 POMDAPI - Environmental Modeling, Optimization and Programming Models

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.

Domaines: Analyse numérique [math.NA]

Eric 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, 2014, 352 (11), pp.941-946. ⟨10.1016/j.crma.2014.09.014⟩. ⟨hal-00994568⟩