Éléments finis mixtes minimaux sur les polyèdres

Given a cellular complex consisting of polytopes, embedded in a Euclidean space, we construct finite element spaces of differential forms, conforming with respect to the exterior derivative, containing those that are polynomial of given maximal degree, having locally the property of exact sequence and extension, so that among all spaces having these properties they have the smallest dimension. More generally we construct, for any finite element system included in a compatible finite element system, an intermediate compatible finite element system of minimal dimension.     

Minorations pour l'équation de Catalan

We consider Catalan's equation x^p − y^q = 1 (with p, q prime and |x|, |y| > 1). We show that, besides the obvious solution 3² − 2³ = 1, min {p; q} > 10⁵ and max {p; q} > 10⁶.     

Méthode des éléments finis multi-échelles pour le problème de Stokes

This work consists of a numerical study of a multi-scale finite element method for a Stokes-type problem with highly oscillating coefficients. The objective of this method is to capture the multi-scale structure of the solution via local basis functions calculated in advance, which contain the essential information on small scales.     

Éléments finis non conformes pour un problème de contact unilatéral avec frottement

In a previous study, we considered it nonconforming finite element method for a frictionless contact problem. The aim of this note is to establish the convergence of the method for a frictional contact problem between two deformable bodies.     

Espaces de Sobolev avec poids pour l'équation de Laplace dans le demi-espace

We deal in this Note with the Laplace equation in the half-space. The use of a special family of weighted Sobolev spaces as a framework is at the heart of our approach. A complete class of existence, uniqueness and regularity results is obtained for inhomogeneous Dirichlet and Neumann problems.     

Approximation d'un problème de l'élasticité non linéaire par une méthode d'éléments finis mixtes

In this Note we introduce a new three field mixed formulation to approximate the solution of a nonlinear elasticity problem based on the Raviart–Thomas type elements. The existence and uniqueness of solutions of the mixed formulation and its discretization are proved and optimal error estimates are derived.     

Passerelles volumes finis — éléments finis

The aim of this work is to introduce a new algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching to a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. The results of numerical experiments concerning a system of two-dimensional, nonlinear partial differential equations on a unstructured mesh are presented.     

Eléments finis nodaux pour les équations de Maxwell

An original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: $\mathbf{E}={\mathbf{E}}^R+\lambda {\mathbf{x}}_P$, where ${\mathbf{E}}^R\in H^1{\left(\omega \right)}^2$, λ depends on the data and domain and ${\mathbf{x}}_P$ is known explicitly. The same splitting can be used for the magnetic field. No cut-off function is needed and improved error estimates are derived. To cite this article: E. Jamelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Méthodes d'éléments finis pour des inéquations variationnelles de contact unilatéral

We give in this Note optimal convergence results for the conforming and non-conforming finite element approximations of the variational inequalities modeling Signorini problem and the unilateral contact between two elastic bodies.     

Sur l'équation de Lamé

Sans résuméExtrait d'une lettre adressée à Mr. Klein     

Éléments finis d'arête et condensation de masse pour les équations de Maxwell: le cas 2D

We construct a new family of triangular and quadrangular edge finite elements for the resolution of Maxwell equations which solves the problem of mass lumping, including the case of anisotropic media. The schemes obtained are analyzed through their numerical dispersion on regular meshes.     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

Équation Périodique de Navier–Stokes sans Viscosité dans une Direction

ABSTRACT

We study the periodic Navier–Stokes equation when the vertical viscosity vanishes, in a critical space (invariant by scaling). We shall prove local-in-time existence of the solution. When the tridimensional part of the initial data is small compared with the bidimesional part and with the horizontal viscosity, we shall show global existence of solutions.     

Solutions faibles d'énergie infinie pour les équations de Navier—Stokes dans ℝ3

We prove existence of suitable weak solutions for the Navier-Stokes equations on (0, ∞) × ℝ³ when the initial value is uniformly locally square integrable and vanishes at infinity.     

Stabilisation Neumann pour l'équation des ondes dans un domaine extérieur

In odd dimension space, we prove, under a microlocal geometric condition, the exponential decay of the local energy for solutions of the wave equation on exterior domains, with Neumann damping ∂_nu+a(x)∂_tu=0.