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 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.

For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.

A new system of integral equations for the exterior 2D time harmonicscattering problem is investigated. This system was first proposed by B. Després in [11]. Two new derivations of this system are given:one from elementary manipulationsof classical equations, the other based on a minimization of a quadratic functional. Numerical issues are addressed to investigate the potentialof the method.

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge.This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

We consider a two dimensional elastic body submitted to unilateral contact conditions, local frictionand adhesion on a part of his boundary. After discretizing the variational formulation with respectto time we use a smoothing technique to approximate the friction term by an auxiliary problem. A shiftingtechnique enables us to obtain the existence of incremental solutions with bounds independent of the regularization parameter. We finally obtain the existence of a quasistatic solution by passing to thelimit with respect to time.

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.

We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop{\rm div}\nolimits\varphi(u,\nabla u)=f$ . The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centeredscheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviartfor the primal unknown u and the div-conformingspace of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the fluxequations.We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.

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.

We study the 3-D elasticity problem in the case of a non-symmetric heterogeneous rod. The asymptotic expansion of the solution is constructed. The coercitivity of the homogenized equation is proved. Estimates are derived for the difference between the truncated series and the exact solution.

This is the second part of the paper for a Non-Newtonian flow. Dual combined Finite Element Methods are used to investigate the littleparameter-dependent problem arising in a nonliner three field version of the Stokes system for incompressible fluids, where the viscosity obeys a general law including the Carreau's law and the Power law. Certain parameter-independent error bounds are obtained which solved the problem proposed by Baranger in [4] in a unifying way. We also give somestable finite element spaces by exemplifying the abstract B-B inequality. The continuous approximation for the extra stress is achieved as a by-product of the new method.

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.

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.

Hermite polynomial interpolation is investigated.Some approximation results are obtained. As an example, the Burgersequation on the whole line is considered. The stability and theconvergence of proposed Hermite pseudospectral scheme are provedstrictly. Numerical results are presented.

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.

In this paper, we study the long wave approximation for quasilinearsymmetric hyperbolic systems. Using the technics developed byJoly-Métivier-Rauch for nonlinear geometrical optics, we prove thatunder suitable assumptions the long wave limit is described byKdV-type systems. The error estimate if the system is coupled appears tobe better. We apply formally our technics to Euler equations with freesurface and Euler-Poisson systems. This leads to new systems of KdV-type.

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.

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 are interested in the theoretical study of a spectral problem arising in a physical situation, namely interactions of fluid-solid type structure. More precisely, we study the existence of solutions for a quadratic eigenvalue problem, which describes the vibrations of a system made up of two elastic bodies, where a slip is allowed on their interface and which surround a cavity full of an inviscid and slightly compressible fluid. The problem shall be treated like a generalized eigenvalue problem. Thus by using some functional analysis results, we deduce the existence of solutions and prove a spectral asymptotic behavior property, which allows us to compare the spectrum of this coupled model and the spectrum associated to the problem without transmission between the fluid-solid media.

We discuss the stability of "critical" or "equilibrium" shapes ofa shape-dependent energy functional. We analyze a problem arising whenlooking at the positivity of the second derivative in order to provethat a critical shape is an optimal shape. Indeed, often whenpositivity -or coercivity- holds, it does for a weaker norm than thenorm for which the functional is twice differentiable and localoptimality cannot be a priori deduced. We solve this problem for aparticular but significant example. We prove "weak-coercivity" ofthe second derivative uniformly in a "strong" neighborhood of theequilibrium shape.