Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials
Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials
View/Date of Publication
08.2016Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11856
Abstract
This paper analyses the numerical solution of a class of non-linear Schrödinger equations by Galerkin finite elements in space and a mass- and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of weak and strong disorder potentials, the consideration of some general class of non-linearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.
Subjects
nonlinear Schrödinger equation, finite elements, Crank–Nicolson, Bose–Einstein condensates
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1142/S0218202517500415Part of Series
INS Preprints ; 1621Henning, Patrick; Peterseim, Daniel: Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials. In: INS Preprints, 1621.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11856
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11856
@unpublished{handle:20.500.11811/11856,
author = {{Patrick Henning} and {Daniel Peterseim}},
title = {Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = aug,
INS Preprints},
volume = 1621,
note = {This paper analyses the numerical solution of a class of non-linear Schrödinger equations by Galerkin finite elements in space and a mass- and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of weak and strong disorder potentials, the consideration of some general class of non-linearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.},
url = {https://hdl.handle.net/20.500.11811/11856}
}
author = {{Patrick Henning} and {Daniel Peterseim}},
title = {Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = aug,
INS Preprints},
volume = 1621,
note = {This paper analyses the numerical solution of a class of non-linear Schrödinger equations by Galerkin finite elements in space and a mass- and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of weak and strong disorder potentials, the consideration of some general class of non-linearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.},
url = {https://hdl.handle.net/20.500.11811/11856}
}
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