We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stableprovided the bifurcation parameter $u_*$ does not exceed a critical value $u_{*}^{c}$ . The latter isthe limit of a decreasing sequence $(u_{*}^{k})$ of bifurcation points. The paper deals with the study of the 2D bifurcated branchesfrom the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we canapply the Crandall-Rabinowitz bifurcation theorem for a local study.We point out that the fully nonlinear reformulation of the FBP can also serve to developefficient numerical schemes in view of global information, such as techniques based on arc length continuation.

An optimal shape control problem for the stationary Navier-Stokessystem is considered. An incompressible, viscous flow in atwo-dimensional channel is studied to determine the shape of part ofthe boundary that minimizes the viscous drag. Theadjoint method and the Lagrangian multiplier method are used to derivethe optimality system for the shapegradient of the design functional.

We investigate sufficient and possiblynecessary conditions for the L 2 stability of the upwind first orderfinite volume scheme for Maxwell equations, with metallic andabsorbing boundary conditions. We yield a very general sufficient condition,valid for any finite volume partition in two and three spacedimensions. We show this condition is necessary for a class ofregular meshes in two space dimensions. However, numerical tests show it is not necessaryin three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numericallyinvestigated.

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.

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.

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently applied in the context of the new schemes.The method is tested on the one-dimensional Euler equations, subject to differentinitial data, and the results are compared to the numericalsolutions, computed by other second-order central schemes.The numerical experiments clearly illustrate the advantages of theproposed technique.

We give an analysis of the stability and uniqueness of the simplylaminated microstructure for all three tetragonal to monoclinicmartensitic transformations. The energy density for tetragonal tomonoclinic transformations has four rotationally invariant wells sincethe transformation has four variants. One of these tetragonal tomonoclinic martensitic transformations corresponds to the shearing ofthe rectangular side, one corresponds to the shearing of the squarebase, and one corresponds to the shearing of the plane orthogonal to adiagonal in the square base. We show that the simply laminatedmicrostructure is stable except for a class of special materialparameters. In each case that the microstructure is stable, we deriveerror estimates for the finite element approximation.

In this paper we present an analysis of the partial differential equations thatdescribe the Chemical Vapor Infiltration (CVI) processes. The mathematical modelrequires at least two partial differential equations, one describing thegas phase and one corresponding to the solid phase.A key difficulty in the process is the long processing times that are typicallyrequired. We address here the issue of optimization and show that we can chooseappropriate pressure and temperature to minimize these processing times.

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.

The behavior of an ordinary differential equation for the low wave number velocitymode is analyzed. This equation was derived in [5]by an iterative process on the two-dimensional Navier-Stokes equations (NSE). Itresembles the NSE in form, exceptthat the kinematic viscosity is replaced by an iterated viscositywhich is a partial sum, dependent on the low-mode velocity. The convergence of this sum as the number of iterations is taken to be arbitrarily large is explored.This leads to a limiting dynamical system which displaysseveral unusual mathematical features.

We consider H(curl;Ω)-elliptic problems that have been discretized bymeans of Nédélec's edge elements on tetrahedral meshes. Suchproblems occur in the numerical computation of eddy currents. From the defectequation we derive localized expressions that can be used as a posteriori error estimators to control adaptiverefinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance for the finite time extinction of the solution.

In this paper, we construct a combined upwinding and mixed finite element method for the numerical solution of a two-dimensional mean field model of superconducting vortices. An advantage of our method is that it works for any unstructured regular triangulation. A simple convergence analysis is given without resorting to the discrete maximum principle. Numerical examples are also presented.

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.

Bhattacharya and Kohn have used small-strain (geometrically linear)elasticity to analyze the recoverable strains of shape-memory polycrystals.The adequacy of small-strain theory is open to question, however, since someshape-memory materials recover as much as 10 percent strain. This paperprovides the first progress toward an analogous geometrically nonlineartheory. We consider a model problem, involving polycrystals madefrom a two-variant elastic material in two space dimensions. The lineartheory predicts that a polycrystal with sufficient symmetry can have norecoverable strain. The nonlinear theory corrects this to the statement thata polycrystal with sufficient symmetry can have recoverable strain nolarger than the 3/2 power of the transformation strain. This result isin a certain sense optimal. Our analysis makes use of Fritz John'stheory of deformations with uniformly small strain.

This work deals with a non linear inverse problem of reconstructingan unknown boundary γ, the boundary conditions prescribed on γ being of Signorini type,by using boundary measurements. The problem is turned into an optimal shape design one, by constructinga Kohn & Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary.Furthermore, we prove that the derivative of this cost function with respect to a direction θ depends only on the state u 0, and not on its Lagrangian derivative u1(θ).

This work is devoted to the study of a two-dimensional vectorPoisson equation with the normal component of the unknown andthe value of the divergence of the unknown prescribed simultaneouslyon the entire boundary.These two scalar boundary conditions appear prima faciealternative in a standard variational framework. An originalvariational formulation of this boundary value problemis proposed here. Furthermore, an uncoupled solution algorithm isintroduced together with its finite element approximation.The numerical scheme has been implemented and appliedto solve a simple test problem.

By means of a result on the semi-global C 1 solution, we establish the exact boundary controllability for the reducible quasilinearhyperbolic system if the C 1 norm of initial data and final state issmall enough.

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.

"Least regret control" consists in trying tofind a control which "optimizes the situation" with the constraint of not making things tooworse with respect to a known reference control,in presence of more or less significantperturbations. This notion was introduced in [7].It is recalled on a simple example (an ellipticsystem, with distributed control and boundary perturbation) in Section 2. We show that the problem reduces to a standard optimal control problem for augmented state equations.On another hand, we have introduced in recentnotes [9-12] the method ofvirtual control, aimed at the"decomposition of everything" (decomposition ofthe domain, of the operator, etc). Anintroduction to this method is presented, withouta priori knowledge needed, in Sections 3 and 4,directly on the augmented state equations.For problems without control, or with "standard" control, numerical applications of the virtualcontrol ideas have been given in the notes[9-12] and in the note[5].One of the first systematic paper devoted to allkind of decomposition methods, including multicriteria, is a jointpaper with A. Bensoussan and R. Temam, towhom this paper is dedicated, cf. [1].