A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations |
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| Authors: | Éric Cancès Geneviève Dusson Yvon Maday Benjamin Stamm Martin Vohralík |
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| Institution: | 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 |
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| Abstract: | 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. |
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