Let $\mathcal{Q}$ be a partition of a polygonal domain of the planinto convexe quadrilaterals. Given a regular function f , we construct afunction πƒ ∈ C2 (Ω), interpolating position values andderivatives of f up of order 2 at vertices of $\mathcal{Q}.$ On eachquadrilateral $Q\in\mathcal{Q},$ πƒ|Q is a finite element obtainedfrom a polynomial scheme of FVS type by adding some rational functions.

We are interested in a barotropic motion of the non-Newtonian bipolarfluids .We consider a specialcase where the stress tensor is expressed in the form ofpotentials depending on e ii and $(\frac{\partiale_{ij}}{\partial x_{k}})$ .We prove theasymptotic stability of the rest state under the assumptionof the regularity of the potential forces.

The LBB condition is well-known to guarantee the stability of a finiteelement (FE) velocity - pressure pair in incompressible flow calculations.To ensure the condition to be satisfied a certain constant should be positive andmesh-independent. The paper studies the dependence of the LBB condition on thedomain geometry. For model domains such as strips and rings thesubstantial dependence of this constant on geometry aspect ratios is observed.In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those whenthe LBB condition is violated. In the core of the paperwe prove that for any FE velocity-pressure pair satisfying usual approximationhypotheses the mesh-independent limit in the LBB condition is not greater thanits continuous counterpart, the constant from the Nečas inequality.For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.

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.

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.

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.

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.

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.

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.

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.

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.