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Un résultat de convergence d'ordre deux en temps pour l'approximation des équations de Navier–Stokes par une technique de projection incrémentale

Jean-Luc Guermond — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The Navier–Stokes equations are approximated by means of a fractional step, Chorin–Temam projection method; the time derivative is approximated by a three-level backward finite difference, whereas the approximation in space is performed by a Galerkin technique. It is shown that the proposed scheme yields an error of 𝒪 ( δ t 2 + h l + 1 ) for the velocity in the norm of (L(Ω)), where ≥ 1 is the polynomial degree of the velocity approximation. It is also shown that the splitting error of projection...

Stabilization of Galerkin approximations of transport equations by subgrid modeling

Jean-Luc Guermond — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

This paper presents a stabilization technique for approximating transport equations. The key idea consists in introducing an artificial diffusion based on a two-level decomposition of the approximation space. The technique is proved to have stability and convergence properties that are similar to that of the streamline diffusion method.

Mixed finite element approximation of an MHD problem involving conducting and insulating regions : the 2D case

Jean Luc GuermondPeter D. Minev — 2002

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

We show that the Maxwell equations in the low frequency limit, in a domain composed of insulating and conducting regions, has a saddle point structure, where the electric field in the insulating region is the Lagrange multiplier that enforces the curl-free constraint on the magnetic field. We propose a mixed finite element technique for solving this problem, and we show that, under mild regularity assumption on the data, Lagrange finite elements can be used as an alternative to edge elements.

Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

Jean-Luc GuermondSerge Prudhomme — 2003

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

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution...

Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

Jean-Luc GuermondSerge Prudhomme — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak...

Evaluation of the condition number in linear systems arising in finite element approximations

Alexandre ErnJean-Luc Guermond — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

This paper derives upper and lower bounds for the p -condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin–Least Squares technique or by means of a non-standard Galerkin technique in ...

Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: the 2D case

Jean Luc GuermondPeter D. Minev — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We show that the Maxwell equations in the low frequency limit, in a domain composed of insulating and conducting regions, has a saddle point structure, where the electric field in the insulating region is the Lagrange multiplier that enforces the curl-free constraint on the magnetic field. We propose a mixed finite element technique for solving this problem, and we show that, under mild regularity assumption on the data, Lagrange finite elements can be used as an alternative to edge elements.

On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions

Jean-Luc GuermondLuigi QuartapelleJiang Zhu — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element approximation. The...

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