Georgios E. Zouraris (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the norm. We prove optimal order error estimates in the and norms, under mild mesh conditions for two and three space dimensions.
Othmar Koch, Christof Neuhauser, Mechthild Thalhammer (2013)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order applied to the multi-configuration...
G.D. Akrivis, V.A. Dougalis, ... (1991)
Numerische Mathematik
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Holger Heumann, Ralf Hiptmair (2013)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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Taking the cue from stabilized Galerkin methods for scalar advection problems, we adapt the technique to boundary value problems modeling the advection of magnetic fields. We provide rigorous error estimates for both fully discontinuous piecewise polynomial trial functions and -conforming finite elements.
Mejjaoli, H. (2009)
Serdica Mathematical Journal
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2000 Mathematics Subject Classification: 35Q55,42B10. In this paper, we study the Schrödinger equation associated with the Dunkl operators, we study the dispersive phenomena and we prove the Strichartz estimates for this equation. Some applications are discussed.
Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik (2014)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank–Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time -stability is proved. Due to the splitting, an effective direct...
Christophe Besse, Brigitte Bidégaray (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.