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 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.

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 treat the problem of constructing exact theories of rods and shells for thin incompressible bodies. We employ a systematic method that consists inimposing constraints to reduce the number of degrees of freedom of each crosssection to a finite number. We show that it is very difficult to produce theories that exactly preserve the incompressibility and we show that it is impossible to do so for naive theories. In particular, many exact theories have nonlocal effects.

This paper deals with a finite element method to solve fluid-structure interaction problems. More precisely it concerns the numerical computation of harmonic hydroelastic vibrations under gravity. It is based on a displacement formulation for both the fluid and the solid. Gravity effects are included on the free surface of the fluid as wellas on the liquid-solid interface. The pressure of the fluid is used as a variable for the theoretical analysis leading to a well posed mixed linear eigenvalue problem.Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a non conforming discretization. The method does not present spurious or circulation modes for nonzero frequencies.Convergence is proved and optimal error estimates are given.Finally, numerical results are shown.

We present the semi-conductor Boltzmann equation,which is time-reversible, and indicate that it can be formally derivedby considering the large time and small perturbing potential limitin the Von-Neumann equation (time-reversible). We then rigorously computethe corresponding asymptotics in the case of the Von-Neumann equation onthe Torus. We show that the limiting equation we obtain does not coincidewith the physically realistic model. The former is indeedan equation of Boltzmann type, yet with memory in time, so that it appearsto be time-reversible. We comment on this point, and further describethe properties of the limiting equation.

Stability and asymptotic stability of the solutions of impulsivenonlinear pa ra bo lic equationsare studied via the method of differential inequalities.

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = α B, where α is a real number.Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper,we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.

The computation of glacier movements leads to a system of nonlinear partial differentialequations. The existence and uniqueness of a weak solution is established by using the calculus ofvariations. A discretization by the finite element method is done. Thesolution of the discrete problem is proved to be convergent to the exactsolution. A first simple numerical algorithm is proposed and its convergence numericallystudied.

We consider space semi-discretizations of the 1-d wave equation in a boundedinterval with homogeneous Dirichlet boundary conditions. We analyze the problemof boundary observability, i.e., the problem of whether the total energy ofsolutions can be estimated uniformly in terms of the energy concentrated on theboundary as the net-spacing h → 0. We prove that, due to the spurious modesthat the numerical scheme introduces at high frequencies, there is no such auniform bound. We prove however a uniform bound in a subspace of solutionsgenerated by the low frequencies of the discrete system. When h → 0 thisfinite-dimensional spaces increase and eventually cover the whole space. Wethus recover the well-known observability property of the continuous systemas the limit of discrete observability estimates as the mesh size tends tozero.We consider both finite-difference and finite-element semi-discretizations.

In this short note we correct a conceptual error in theheuristic derivation of a kinetic equation used for thedescription of a one-dimensional granular medium in the socalled quasi-elastic limit, presented by the same authors inreference[1]. The equation we derived is however correct so that,the rigorous analysis on this equation, which constituted themain purpose of that paper, remains unchanged.