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 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 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.

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.

We consider the numerical approximation of a first orderstationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh ${\cal T}_h$ . For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem $P_h^k$ is well posed and weobtain an error estimate. We show that when k goes to zero problem $(P_h^k)$ (resp. the || ||h,k norm)has as a limit problem (P h ) (resp. the || ||h norm) associated to the Galerkin discontinuousmethod. This extends to two and three space dimension our previous results obtained in one space dimension.

The fluctuation splitting schemes were introduced by Roe in the beginning of the80's and have been then developed since then, essentially thanks to Deconinck.In this paper, the fluctuation splittingschemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of thethree dimensional Euler equations is presented. This decomposition leads to an acousticsubsystem and two scalar advection equations, one of them being the entropy advection.Thanks to this decomposition, the two scalar equations are treated with the well knownPSI scalar fluctuation splitting scheme, and the acoustic subsystem is treatedwith the Lax Wendroff matrix fluctuation splitting scheme. Anadditional viscous term is introduced in order to reduce the oscillatory behavior of theLax Wendroff scheme. An implicit form leadsto a robust scheme which enables computations over a large rangeof Mach number. This fluctuation splitting scheme, called the Lax Wendroff - PSI scheme, produceslittlespurious entropy, thus leading to accurate drag predictions.Numerical results obtained with this Lax Wendroff PSI scheme are presented and compared to areference Euler code, based on a Lax Wendroff scheme.

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.