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 convergenceproperties that are similar to that of the streamline diffusion method.

We consider the numerical approximation of a first orderstationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh ${\cal T}_h$ . For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem $P_h^k$ is well posed and weobtain an error estimate. We show that when k goes to zero problem $(P_h^k)$ (resp. the || ||h,k norm)has as a limit problem (P h ) (resp. the || ||h norm) associated to the Galerkin discontinuousmethod. This extends to two and three space dimension our previous results obtained in one space dimension.

The fluctuation splitting schemes were introduced by Roe in the beginning of the80's and have been then developed since then, essentially thanks to Deconinck.In this paper, the fluctuation splittingschemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of thethree dimensional Euler equations is presented. This decomposition leads to an acousticsubsystem and two scalar advection equations, one of them being the entropy advection.Thanks to this decomposition, the two scalar equations are treated with the well knownPSI scalar fluctuation splitting scheme, and the acoustic subsystem is treatedwith the Lax Wendroff matrix fluctuation splitting scheme. Anadditional viscous term is introduced in order to reduce the oscillatory behavior of theLax Wendroff scheme. An implicit form leadsto a robust scheme which enables computations over a large rangeof Mach number. This fluctuation splitting scheme, called the Lax Wendroff - PSI scheme, produceslittlespurious entropy, thus leading to accurate drag predictions.Numerical results obtained with this Lax Wendroff PSI scheme are presented and compared to areference Euler code, based on a Lax Wendroff scheme.

We study the asymptotic behaviour of the following nonlinear problem: $$\{\begin{array}{ll}-{\rm div}(a( Du_h))+\vert u_h\vert^{p-2}u_h =f \quad\hbox{in }\Omega_h, a( Du_h)\cdot\nu = 0 \quad\hbox{on }\partial\Omega_h, \end{array}.$$

in a domain Ωh of $\mathbb{R}^n$ whose boundary ∂Ωh contains an oscillating part with respect to hwhen h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h -1-periodically distributed. We prove that the limit problem in the domain corresponding tothe oscillating boundary identifieswith a diffusion operator with respect tox n coupled with an algebraic problemfor the limit fluxes.

The standard discretization of the Stokes andNavier–Stokes equations in vorticity and stream function formulation by affine finiteelements is known for its bad convergence. We present here a modified discretization, weprove that the convergence is improved and we establish a priori error estimates.

We consider the analysis and numerical solution of a forward-backward boundary value problem.We provide some motivation, prove existence and uniqueness in a functionclass especially geared to the problem at hand, provide various energyestimates, prove a priori error estimates for the Galerkin method,and show the results of some numerical computations.

A justification of the two-dimensional nonlinear “membrane”equations for a plate made of a Saint Venant-Kirchhoff material hasbeen given by Fox et al. [9] by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the originalthree dimensional problem in the sense that its energy density isinvariant under the rotations of ${\mathbb{R}}^3$ , is equivalent to finding thecritical points of a functional whose nonlinear part depends on the firstfundamental form of the unknown deformed surface. We establish here an existence result for these equations in the case of themembrane submitted to a boundary condition of “tension”, and we show that thesolution found in our analysis is injective and is the unique minimizer of thenonlinear membrane functional, which is not sequentially weakly lowersemi-continuous.We also analyze the behaviour of the membrane when the “tension” goes toinfinityand we conclude that a “well-extended” membrane may undergo largeloadings.

In the case of an elastic strip we exhibit two properties ofdispersion curves λn,n ≥ 1, that were not pointed outpreviously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on ${\mathbb{R}}_{+}$ . The non monotonicity was an open question (see [2],for example) and, for the first time, we give a rigourous answer. Recall thecharacteristic property of the dispersion curves: {λn(p);n ≥ 1} isthe set of eigenvalues of A p , counted with their multiplicity. Theoperators Ap , $p\in{\mathbb{R}}$ , are the reduced operators deduced from the elasticoperator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relationD(p,λ) = 0 in a general framework, and not only for a homogeneous situation(in this last case the relation is explicit). Recall that a dispersionrelation isan implicit equation the solutions of which are eigenvalues of A p . The mainproperty of the function D that we build is the following one: themultiplicity of an eigenvalue λ of A p is equal to the multiplicity ithas as a root of D(p,λ) = 0. We give also some applications.

In this paper, we consider second order neutrons diffusion problem withcoefficients in L ∞(Ω). Nodal method of the lowest order is applied to approximate the problem's solution. The approximation uses special basis functions [1] in which the coefficientsappear. The rate of convergence obtained is O(h2) in L 2(Ω), with a free rectangular triangulation.

Using the approach in [5] for analysingtime discretization error and assuming more regularity on the initial data, we improve onthe error bound derived in [2] for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy withnon-smooth free energy.

In this paper we study the derivation of homogeneous hydrostatic equationsstarting from 2D Euler equations, following for instance[2,9]. We give a convergence result for convex profiles anda divergence result for a particular inflexion profile.

Based on QR-like decomposition with column pivoting, a new and efficient numerical method for solving symmetric matrix inverse eigenvalue problems is proposed, which is suitable for both the distinct and multiple eigenvalue cases. A locally quadratic convergence analysis is given. Some numerical experiments are presented to illustrate our results.

We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ 1, Φ 2, Φ 3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ 1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H ∑ of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions.

We consider a finite element discretization bythe Taylor–Hood element for the stationaryStokes and Navier–Stokesequations with slip boundary condition. The slip boundary conditionis enforced pointwise for nodal values of the velocity inboundary nodes. We prove optimal error estimates in theH 1 and L 2 norms for the velocity and pressure respectively.

We obtain, for entire functions of exponential type satisfying certainintegrability conditions, a quadrature formula using the zeros of sphericalBessel functions as nodes. We deduce from this quadrature formula aresult of Olivier and Rahman, which refines itself a formula of Boas.

For domains which are star-shapedw.r.t. at least one point, we give new bounds on theconstants in Jackson-inequalities in Sobolev spaces. Forconvex domains, these bounds do not depend on theeccentricity. For non-convex domains with a re-entrantcorner, the bounds are uniform w.r.t. the exteriorangle. The main tool is a new projection operator ontothe space of polynomials.

Nous écrivons et nous justifions des conditions aux limites approchéespour descouches minces périodiques recouvrant un objetparfaitement conducteur en polarisation transverse électrique ettransverse magnétique.

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations.

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of vergence by a sequence ofintegral functionals defined on piecewise affine functions.