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Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Nicolas Besse, Dietmar Kröner (2005)

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

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition Δ t h 4 / 3 , we obtain error estimates in L 2 of order 𝒪 ( Δ t 2 + h m + 1 / 2 ) where m is the degree of the local polynomials.

Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Nicolas Besse, Dietmar Kröner (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition Δ t h 4 / 3 , we obtain error estimates in L2 of order 𝒪 ( Δ t 2 + h m + 1 / 2 ) where m is the degree of the local polynomials.

Convergence of Rothe's method in Hölder spaces

Norio Kikuchi, Jozef Kačur (2003)

Applications of Mathematics

The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.

Convergence of the finite element method applied to an anisotropic phase-field model

Erik Burman, Daniel Kessler, Jacques Rappaz (2004)

Annales mathématiques Blaise Pascal

We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H 1 -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not...

Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Béla J. Szekeres, Ferenc Izsák (2017)

Applications of Mathematics

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated...

Convergence Rates of the POD–Greedy Method

Bernard Haasdonk (2013)

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

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy...

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

Eliane Bécache, Jeronimo Rodríguez, Chrysoula Tsogka (2009)

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

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

Eliane Bécache, Jeronimo Rodríguez, Chrysoula Tsogka (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...

Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system

Andreas Prohl, Markus Schmuck (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system

Andreas Prohl (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain Ω 3 . We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity...

Convergent semidiscretization of a nonlinear fourth order parabolic system

Ansgar Jüngel, René Pinnau (2003)

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

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

Convergent semidiscretization of a nonlinear fourth order parabolic system

Ansgar Jüngel, René Pinnau (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

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