Let $\mathcal{Q}$ be a partition of a polygonal domain of the planinto convexe quadrilaterals. Given a regular function f , we construct afunction πƒ ∈ C2 (Ω), interpolating position values andderivatives of f up of order 2 at vertices of $\mathcal{Q}.$ On eachquadrilateral $Q\in\mathcal{Q},$ πƒ|Q is a finite element obtainedfrom a polynomial scheme of FVS type by adding some rational functions.

The aim of this work is to introduce and to analyze new algorithms for solving the transport neutronique equation in 2D geometry. These algorithms present the duplicate favors to be, on the one hand faster than some classic algorithms and easily to be implemented and naturally deviced for parallelisation on the other hand. They are based on a splitting of the collision operator holding amount of caracteristics of the transport operator. Some numerical results are given at the end of this work.

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theoryas the relative thickness ε of the shell tends tozero. For ε = 0 our problem is parabolic, then it is amodel of developpable surfaces. Boundary layers along and across the characteristichave very different structure. It also appears internal layers associatedwith propagations of singularities along the characteristics. The specialstructure of the limit problem often implies solutions which exhibitdistributional singularities along the characteristics. The correspondinglayers for small ε have a very large intensity. Layers alongthe characteristics have a special structure involving subspaces; thecorresponding Lagrange multipliers are exhibited. Numerical experimentsshow the advantage of adaptive meshes in these problems.

The hydrostatic approximation of the incompressible 3D stationaryNavier-Stokes equations is widely used in oceanography and other applied sciences. It appears through a limit process due to the anisotropy of the domain in use, an ocean, and it is usually studied as such.We consider in this paper an equivalent formulation to this hydrostatic approximation that includes Coriolis force and an additional pressure term that comes from taking into account the pressure in the state equation for the density. It therefore models a slight dependence of the density upon compression terms. We study this model as an independent mathematical object and prove an existence theorem by means of a mixed variational formulation. The proof uses a family of finite element spaces to discretize the problem coupled with a limit process that yields the solution.We finish this paper with an existence and uniqueness result for the evolutionary linear problem associated to this model. This problem includes the same additional pressure term and Coriolis force.

We consider solutions to the time-harmonic Maxwell's Equationsof a TE (transverse electric) nature. For such solutions we providea rigorous derivation of the leading order boundary perturbationsresulting from the presence of a finite number of interior inhomogeneitiesof small diameter. We expect that these formulas will form the basis for very effective computational identification algorithms, aimed at determininginformation about the inhomogeneities from electromagnetic boundary measurements.

We derive in this article some models ofCahn-Hilliard equations in nonisotropic media. These models, based onconstitutive equations introduced by Gurtin in [19], take the work ofinternal microforces and also the deformations of the material intoaccount. We then study the existence and uniqueness of solutions andobtain the existence of finite dimensional attractors.

Fast singular oscillating limits ofthe three-dimensional "primitive" equations ofgeophysical fluid flows are analyzed.We prove existence on infinite time intervals of regular solutions to the3D "primitive" Navier-Stokes equations for strongstratification (large stratification parameter N).This uniform existence is proven forperiodic or stress-free boundary conditionsfor all domain aspect ratios,including the case of three wave resonances which yield nonlinear " $2\frac{1}{2}$ dimensional" limit equations for N → +∞;smoothness assumptions are the same as for localexistence theorems, that is initial data in Hα , α ≥ 3/4.The global existence is proven using techniques ofthe Littlewood-Paley dyadic decomposition.Infinite time regularity for solutions of the3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonantequations and convergence theorems.

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h).

The present work is a mathematical analysis of two algorithms, namelythe Roothaan and the level-shifting algorithms, commonly used inpractice to solve the Hartree-Fock equations. The level-shiftingalgorithm is proved to be well-posed and to converge provided the shiftparameter is large enough. On the contrary, cases when the Roothaanalgorithm is not well defined or fails in converging areexhibited. These mathematical results are confronted to numericalexperiments performed by chemists.

We are interested in a barotropic motion of the non-Newtonian bipolarfluids .We consider a specialcase where the stress tensor is expressed in the form ofpotentials depending on e ii and $(\frac{\partiale_{ij}}{\partial x_{k}})$ .We prove theasymptotic stability of the rest state under the assumptionof the regularity of the potential forces.

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

In this article we consider local solutions for stochastic Navier Stokesequations, based on the approach of Von Wahl, for the deterministic case. Wepresent several approaches of the concept, depending on the smoothnessavailable. When smoothness is available, we can in someway reduce thestochastic equation to a deterministic one with a random parameter. In thegeneral case, we mimic the concept of local solution for stochasticdifferential equations.

We consider a domain decomposition method for some unsteadyheat conduction problem in composite structures.This linear model problem is obtained by homogenization of thin layersof fibres embedded into some standard material.For ease of presentation we consider the case of two space dimensions only.The set of finite element equations obtained by the backward Euler schemeis parallelized in a problem-oriented fashion by some noniterative overlappingdomain splitting method,eventually enhanced by inexpensive local iterationsto reduce the overlap.We present a detailed convergence analysis of this algorithmwhich is particularly well appropriate to handle fibre layersof nonlinear material.Special emphasis is to take into account the specific regularity propertiesof the present mathematical model.Numerical experiments show the reliability of the theoretical predictions.

Consider the domain $Z_\epsilon=\{x\in\mathbb{R}^n ; {dist}(x,\epsilon\mathbb{Z}^n)> \epsilon^\gamma\}$ and let the free path length be defined as $\tau_\epsilon(x,v)=\inf\{t> 0 ; x-tv\in Z_\epsilon\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma=\frac{n}{n-1}$ , it is shownthat the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon$ is bounded from belowby an expression of the form C/t, for some C> 0. A numerical study seems toindicate that asymptotically for large t, $\phi_\epsilon\sim C/t$ .This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As aconsequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describethe Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usualcase of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].

The paper is devoted to analysis of an elliptic-algebraic system ofequations describing heat explosion in a two phase medium filling a star-shaped domain. Three typesof solutions are found: classical, critical andmultivalued. Regularity of solutions is studied as well as theirbehavior depending on the size of the domain and on the coefficient ofheat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

The LBB condition is well-known to guarantee the stability of a finiteelement (FE) velocity - pressure pair in incompressible flow calculations.To ensure the condition to be satisfied a certain constant should be positive andmesh-independent. The paper studies the dependence of the LBB condition on thedomain geometry. For model domains such as strips and rings thesubstantial dependence of this constant on geometry aspect ratios is observed.In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those whenthe LBB condition is violated. In the core of the paperwe prove that for any FE velocity-pressure pair satisfying usual approximationhypotheses the mesh-independent limit in the LBB condition is not greater thanits continuous counterpart, the constant from the Nečas inequality.For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperatureθ and an ODE with double obstaclesfor phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requiresthe stability constraint τ ≤ ε. Conforming piecewiselinear finite elements over highly graded simplicial mesheswith parameter h are further employed for space discretization.A posteriori errorestimates are derived for both unknowns θ and χ, whichexhibit the correct asymptotic order in terms of ε, h andτ. This result circumvents the use of duality, which does noteven apply in this context.Several numerical experiments illustrate the reliability of theestimators and document the excellent performance of the ensuingadaptive method.

In dimension one it is proved that the solution to a total variation-regularizedleast-squares problem is always a function which is "constant almost everywhere" ,provided that the data are in a certain sense outside the range of the operatorto be inverted. A similar, but weaker result is derived in dimension two.

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs , with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.