For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.

The paper is devoted to the convergence analysis of a well-knowncell-centered Finite Volume Method (FVM) for aconvection-diffusion problem in $\mathbb{R}^2$ . This FVM is based on Voronoi boxes andexponential fitting. To prove the convergence of the FVM, we usea new nonconforming Petrov-Galerkin Finite Element Method (FEM)for which the system of linear equations coincides completely withthat of the FVM. Thus, by proving convergence properties of theFEM we obtain similar ones for the FVM. For the error estimationof the FEM well-known statements have to be modified.

Evolution equations featuring nonlinearity, dispersion anddissipation are considered here. For classes of such equationsthat include the Korteweg-de Vries-Burgers equation and the BBM-Burgers equation, the zero dissipation limit is studied.Uniform bounds independent of the dissipation coefficient are derived and zero dissipation limit results with optimal convergence rates are established.

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 two dimensional elastic body submitted to unilateral contact conditions, local frictionand adhesion on a part of his boundary. After discretizing the variational formulation with respectto time we use a smoothing technique to approximate the friction term by an auxiliary problem. A shiftingtechnique enables us to obtain the existence of incremental solutions with bounds independent of the regularization parameter. We finally obtain the existence of a quasistatic solution by passing to thelimit with respect to time.

We present an improved, near-optimal hp error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new hp mortaring technique, called the mortar method (MP), and derive h, p and hp error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the hp method.

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problemssuffers under a coarse boundary approximation. To overcome this drawback the methodis furnished withan a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme.In contrast to similar, but rather theoretical, concepts already described in theliterature our approach combines a high generality with an easy-to-implement algorithm.

Non reflecting boundary conditions on artificial frontiersof the domain are proposed for bothincompressible and compressible Navier-Stokes equations.For incompressible flows, the boundary conditions lead to a well-posedproblem, convey properly the vortices without any reflections on theartificial limits and allow to compute turbulent flows at high Reynoldsnumbers.For compressible flows, the boundary conditions convey properly thevortices without any reflections on the artificial limits and alsoavoid acoustic waves that go back into the flow and change itsbehaviour.Numerical tests illustrate the efficiency of the various boundaryconditions.

We discuss the stability of "critical" or "equilibrium" shapes ofa shape-dependent energy functional. We analyze a problem arising whenlooking at the positivity of the second derivative in order to provethat a critical shape is an optimal shape. Indeed, often whenpositivity -or coercivity- holds, it does for a weaker norm than thenorm for which the functional is twice differentiable and localoptimality cannot be a priori deduced. We solve this problem for aparticular but significant example. We prove "weak-coercivity" ofthe second derivative uniformly in a "strong" neighborhood of theequilibrium shape.

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.

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 are interested in the theoretical study of a spectral problem arising in a physical situation, namely interactions of fluid-solid type structure. More precisely, we study the existence of solutions for a quadratic eigenvalue problem, which describes the vibrations of a system made up of two elastic bodies, where a slip is allowed on their interface and which surround a cavity full of an inviscid and slightly compressible fluid. The problem shall be treated like a generalized eigenvalue problem. Thus by using some functional analysis results, we deduce the existence of solutions and prove a spectral asymptotic behavior property, which allows us to compare the spectrum of this coupled model and the spectrum associated to the problem without transmission between the fluid-solid media.

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge.This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

We study the well-posedness of an unsteady fluid-structure interaction problem.We consider a viscous incompressible flow, which is modelled by theNavier-Stokes equations. The structure is a collection of rigid moving bodies. The fluiddomain depends on time and is defined by the position of the structure, itself resultingfrom a stress distribution coming from the fluid. The problem is thennonlinear and the equations we deal with are coupled. We prove its localsolvability in time through two fixed point procedures.

In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from anassociated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

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.

Hermite polynomial interpolation is investigated.Some approximation results are obtained. As an example, the Burgersequation on the whole line is considered. The stability and theconvergence of proposed Hermite pseudospectral scheme are provedstrictly. Numerical results are presented.

A Legendre spectral collocation method is presented for the solutionof the biharmonic Dirichlet problem on a square. The solution andits Laplacian are approximated using the set of basis functions suggestedby Shen, which are linear combinations of Legendre polynomials. A Schurcomplement approach is used to reduce the resulting linear system to oneinvolving the approximation of the Laplacian of the solution on the twovertical sides of the square. The Schur complement system is solved bya preconditioned conjugate gradient method. The total cost of the algorithmis O(N 3). Numerical results demonstrate the spectral convergence of the method.

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.

A new system of integral equations for the exterior 2D time harmonicscattering problem is investigated. This system was first proposed by B. Després in [11]. Two new derivations of this system are given:one from elementary manipulationsof classical equations, the other based on a minimization of a quadratic functional. Numerical issues are addressed to investigate the potentialof the method.