The Mumford-Shah functional for image segmentation is an original approachof the image segmentation problem, based on a minimal energy criterion. Itsminimization can be seen as a free discontinuity problem and is based onΓ-convergence and bounded variation functions theories. Some newregularization results, make possible to imagine a finite element resolutionmethod. In a first time, the Mumford-Shah functional isintroduced and some existing results are quoted. Then, adiscrete formulation for the Mumford-Shah problem is proposedand its Γ-convergence is proved. Finally, somenumerical results, computed from both artificial and realimages are presented and discussed.

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.

Lagrangian and augmented Lagrangian methods for nondifferentiableoptimization problems that arise from the total bounded variation formulation of image restoration problems are analyzed. Conditional convergence of theUzawa algorithm and unconditional convergence of the first order augmentedLagrangian schemes are discussed. A Newton type method based on an activeset strategy defined by means of the dual variables is developed andanalyzed. Numerical examples for blocky signals and images perturbedby very high noise are included.

This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^{1/2}$ , studied by Amrouche and Cioranescu, the H 1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to the general case of third grade, complicates substantially the proof of the existence of the solution. We also prove further regularity results by a method similar to that of Cioranescu and Girault for second grade fluids. This extension to the third grade fluids is not straightforward, because of a transport equation which is much more complex.

We present in this paper a stability study concerning finite volume schemesapplied to the two-dimensional Maxwell system, using rectangular or triangular meshes. A stability condition is proved for thefirst-order upwind scheme on a rectangular mesh. Stability comparisonsbetween the Yee scheme and the finite volume formulation are proposed.We also compare the stability domains obtained when considering theMaxwell system and the convection equation.

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.

In this paper we introduce a coupled systems of kinetic equations forthe linearized Carleman model. We then study the existencetheory andthe asymptotic behaviour of the resulting coupled problem. In order tosolve the coupled problem we propose to use the time marching algorithm.We then develop a convergence theory for the resulting algorithm. Numericalresults confirming the theory are then presented.

We discuss a parallel implementation of the domaindecomposition method based on the macro-hybrid formulationof a second order ellipticequation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonalpreconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.

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

We propose and analyse a abstract framework for augmented mixed formulations.We give a priori error estimate in the general case: conforming and nonconforming approximations with or without numerical integration.Finally, a posteriori error estimator is given. An example of stabilized formulation for Stokes problem is analysed.

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.

In this paper we consider the numerical computation of the optimal costfunction associated to the problem that consists in finding the minimum ofthe maximum of a scalar functional on a trajectory. We present anapproximation method for the numerical solution which employs bothdiscretization on time and on spatial variables. In this way, we obtain afully discrete problem that has unique solution. We give an optimal estimatefor the error between the approximated solution and the optimal costfunction of the original problem. Also, numerical examples are presented.

We propose a two point subdivision scheme with parameters to draw curves that satisfy Hermite conditions at both ends of [a,b]. We build three functions f,p and s on dyadic numbers and, using infinite products of matrices, we prove that, under assumptions on the parameters, these functions can be extended by continuity on [a,b], with f'=p and f''=s .

The idea of replacing a divergence constraint by a divergence boundarycondition is investigated. The connections between the formulations areconsidered in detail. It is shown that the most common methods of usingdivergence boundary conditions do not always work properly. Necessaryand sufficient conditions for the equivalence of the formulations aregiven.

We derive explicit bounds on the constantsin error estimates for two quasi-interpolation operators which aremodifications of the “classical” Clément-operator.These estimates are crucial for making explicit theconstants which appear in popular a posteriori errorestimates. They are also compared with correspondingestimates for the standard nodal interpolation operator.

About two years ago, Gobbino [21]gave a proof of a De Giorgi's conjectureon the approximation of the Mumford-Shah energy by means offinite-differences based non-local functionals.In this work, we introduce a discretized version of De Giorgi'sapproximation, that may be seen as a generalization ofBlake and Zisserman's “weak membrane” energy(first introduced in the image segmentation framework).A simple adaptation of Gobbino's results allows us tocompute the Γ-limit of this discrete functional asthe discretization step goes to zero; this generalizes a previouswork by the author on the “weak membrane” model [10].We deduce how to design in a systematic way discreteimage segmentation functionals with “less anisotropy” thanBlake and Zisserman's original energy, and we show insome numerical experiments how it improves the method.

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 consider plasma tearing mode instabilities when the resistivity depends on a flux function (ψ), for the plane slab model.This problem, represented by the MHD equations, is studied as a bifurcation problem. Forso doing, it is written in the form (I(.)-T(S,.)) = 0, whereT(S,.) is a compact operator in a suitable space and S is the bifurcationparameter.In this work, the resistivity is not assumed to be a given quantity (as usuallydone in previous papers, see [1,2,5,7,8,9,10], but it depends nonlinearly of the unknowns of the problem; this is the main difficulty, with newmathematical results.We also develop in this paper a 1D code to compute bifurcation points from the trivialbranch (equilibrium state).

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate oforder h, assuming only the W2,p (for p>2) regularity of thecontinuous solution, on a mesh of quadrangles. The proof is based on anextension of the ideas developed in [12]. Some newdifficulties arise here, due to the weak regularity of the solution, and thenecessity to approximate the entire gradient, and not only its normalcomponent, as in [12].

We give results for the approximation of a laminate withvarying volume fractions for multi-well energy minimizationproblems modeling martensitic crystals thatcan undergo either an orthorhombicto monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfythe corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energyfor the approximation of the limiting macroscopicdeformation and the simply laminated microstructure.Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.