On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation

Georgios E. Zouraris

Displaying similar documents to “On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation”

Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

Marian Slodička (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $Ω \subset ℝ^{dim}$ with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part $Γ_{n}$ . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the...

Discrete Sobolev inequalities and $L^{p}$ error estimates for finite volume solutions of convection diffusion equations

Yves Coudière, Thierry Gallouët, Raphaèle Herbin (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce $L^{p}$ error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

On the convergence of a factorized vector finite difference scheme.

Jovanović, Boško (1996)

Publications de l'Institut Mathématique. Nouvelle Série

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Numerical study of self-focusing solutions to the Schrödinger-Debye system

Christophe Besse, Brigitte Bidégaray (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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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.

On some nonlinear potential problems.

Efendiev, M.A., Schmitz, Hermann, Wendland, Wolfgang L. (1999)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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Two-scale FEM for homogenization problems

Ana-Maria Matache, Christoph Schwab (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale $ε ≪ 1$ is analyzed. Full elliptic regularity independent of $ε$ is shown when the solution is viewed as mapping from the slow into the fast scale. Two-scale FE spaces which are able to resolve the $ε$ scale of the solution with work independent of $ε$ and without analytical homogenization are introduced. Robust in $ε$ error estimates for the two-scale FE spaces...

On fully practical finite element approximations of degenerate Cahn-Hilliard systems

John W. Barrett, James F. Blowey, Harald Garcke (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments...

Numerical solution of parabolic equations in high dimensions

Tobias Von Petersdorff, Christoph Schwab (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We consider the numerical solution of diffusion problems in $(0, T) \times Ω$ for $Ω \subset ℝ^{d}$ and for $T > 0$ in dimension $d \geq 1$ . We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p \geq 1$ , and $h p$ discontinuous Galerkin time-discretization of order $r = O (|log h|)$ on a geometric sequence of $O (|log h|)$ many time steps. The linear systems in each time step are solved iteratively by $O (|log h|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an $L^{2} (Ω)$ -error of $O (N^{- p})$ for $u (x, T)$ where $N$ is the total number of...

$L^{2}$ -error estimates of the extrapolated Crank-Nicolson discontinuous Galerkin approximations for nonlinear Sobolev equations.

Ohm, Mi Ray, Lee, Hyun Young, Shin, Jun Yong (2010)

Journal of Inequalities and Applications [electronic only]

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On the convergence of multicomponent three level alternating direction difference scheme.

Jovanović, Boško S. (1994)

Matematichki Vesnik

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Linearization of nonlinear differential equations. IV: Nonlinear second order partial differential equations equivalent to linear base equation.

Kocić, Vlajko Lj. (1983)

Publications de l'Institut Mathématique. Nouvelle Série

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Convergence of a Finite Difference Method for the Third BVP for Poisson's Equation

Boško Jovanović, Branislav Popović (1999)

Matematički Vesnik

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