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.

Motivated by rotating fluids, we study incompressible fluidswith anisotropic viscosity.We use anisotropic spaces that enable us to prove existencetheoremsfor less regular initial data than usual. In the case of rotatingfluids, in the whole space, we prove Strichartz-type anisotropic, dispersive estimateswhich allow us to prove global wellposedness for fast enoughrotation.

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.

In the shape from shading problem of computer vision oneattempts to recover the three-dimensional shape of an object orlandscape from the shading on a single image. Under theassumptions that the surface is dusty, distant, and illuminatedonly from above, the problem reduces to that of solving theeikonal equation |Du|=f on a domain in $\mathbb{R}^2$ . Despitevarious existence and uniqueness theorems for smooth solutions,we show that this problem is unstable, which is catastrophic forgeneral numerical algorithms.

Finite element semidiscrete approximations on nonlinear dynamicshallow shell models in considered. It is shown that the algorithmleads to global, optimal rates of convergence. The resultpresented in the paper improves upon the existing literature where therates of convergence were derived for small initial data only[19].

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theoryas the relative thickness ε of the shell tends tozero. For ε = 0 our problem is parabolic, then it is amodel of developpable surfaces. Boundary layers along and across the characteristichave very different structure. It also appears internal layers associatedwith propagations of singularities along the characteristics. The specialstructure of the limit problem often implies solutions which exhibitdistributional singularities along the characteristics. The correspondinglayers for small ε have a very large intensity. Layers alongthe characteristics have a special structure involving subspaces; thecorresponding Lagrange multipliers are exhibited. Numerical experimentsshow the advantage of adaptive meshes in these problems.

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.

In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such thatfor the extended pair the Babuska-Brezzi condition is satisfied. Examplesof stable elements which can be derived in such a way imply the stability of the well-knownQ2/Q1 -element and the 4Q1/Q1 -element. However, our new elementsare much more cheaper. In particular, we shall see that more than half of theadditional degrees of freedom when switching from the Q 1 to the Q 2 and4Q1 , respectively, element are not necessary to stabilize theQ1/Q1 -element. Moreover, by using the technique of reduced discretizationsand eliminating the additional degrees of freedom we showthe relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilizedmethods for the Stokes and Navier-Stokes equations. Finally, we showhow the Brezzi-Pitkäranta stabilization and the SUPG method for theincompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

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.

We consider a domain decomposition method for some unsteadyheat conduction problem in composite structures.This linear model problem is obtained by homogenization of thin layersof fibres embedded into some standard material.For ease of presentation we consider the case of two space dimensions only.The set of finite element equations obtained by the backward Euler schemeis parallelized in a problem-oriented fashion by some noniterative overlappingdomain splitting method,eventually enhanced by inexpensive local iterationsto reduce the overlap.We present a detailed convergence analysis of this algorithmwhich is particularly well appropriate to handle fibre layersof nonlinear material.Special emphasis is to take into account the specific regularity propertiesof the present mathematical model.Numerical experiments show the reliability of the theoretical predictions.

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

The aim of this work is to establish, from amathematical point of view, the limit α → +∞ in the system $i \partial_t E+\nabla (\nabla . E)-\alpha^2 \nabla \times\nabla \times E =-|E|^{2\sigma}E,$ where $E:{\ensuremath{{\Bbb R}}}^3\rightarrow{\mathbb C}^3$ . This corresponds to an approximationwhich is made in the context of Langmuir turbulence in plasmaPhysics. The L 2-subcritical σ (that is σ ≤ 2/3)and the H 1-subcritical σ (that is σ ≤ 2) arestudied. In the physical case σ = 1, the limit is then studied for the $H^1({\ensuremath{{\Bbb R}}}^3)$ norm.

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, several modifications of the quasi-interpolationoperator of Scott andZhang [30]are discussed. The modified operators are defined for non-smooth functionsand are suited for application on anisotropic meshes. Theanisotropy of the elements is reflected in the local stability andapproximation error estimates.As an application, an example is considered where anisotropic finiteelement meshes are appropriate, namely the Poisson problem in domainswith edges.

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.

One of the main tools in the proof of residual-based a posteriori error estimates is a quasi-interpolation operator due to Clément. We modify this operator in the setting of a partition of unitywith the effect that the approximation error has a local average zero.This results in a new residual-based a posteriori error estimate with a volume contribution which is smaller than in the standard estimate.For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed finite element methods.

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfacesin natural norms. Furthermore we prove a global existence result for thecontinuous problem and present some examples of computed surfaces.

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.

In this paper, a phase field system of Penrose–Fife type withnon–conserved order parameter is considered. A class of time–discrete schemes for an initial–boundary value problem for this phase–field system is presented.In three space dimensions, convergence is proved and an error estimate linear with respect to the time–step sizeis derived.

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.