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In this paper a numerical analysis of in situ-biorestoration is presented. Improved stability and error estimates are derived for a P 1 finite-element method applied to coupled system of non linear partial differential equations modeling flow in porous media. These results improve upon previously derived stability and error estimates in two respects: first, a stability of the approximate solution is demonstrated in the case of a non linear diffusive flux λ ( u ) ∇ u with weaker norm assumptions than before and depends only on the initial conditions and time interval, and second, error estimates are optimal as in linear case. Extensions include the finite element approximation of flow field, described by Darcy’s law under the stream function–vorticity formulation because of its multiple advantages. Finally, numerical simulations for a correlation exponential plume are presented.

This article offers a mathematical model that is in line with three-dimensional informet- rics. We show that the geometric distribution – often found in informetric processes of use to model the distributions of book loans in libraries, of citations, or of journals downloads in scientific databases – plays a surprising role. To show this, we infer the existence of another distribution, which we have called “distribution of structure”. Distributions of use and distributions of structure are the two key processes in our three-dimensional IPP (Infor- mation Production Process). Furthermore, we indirectly introduce the time dependence of these distributions through the concept of information obsolescence. To demonstrate the theorems, we use the properties of probability-generating functions (PGF).

We introduce and analyze a simple nonconforming quadrilateral finite element and then we derive optimal a priori error estimates for arbitrary regular quadrilaterals. The idea of extension to some non-conforming elements for three-dimensional hexahedrons is also presented.

This short note is devoted to the proof of the L q -saturation property for functions from the Sobolev space W 2 , p ( Ω ) , p > d / 2 , on sequences of conformal, possibly anisotropic, simplicial meshes for spaces of arbitrary dimension d . The proof completes the theory of optimal meshes minimizing the L q -error ( 0 q ≤ p ) of P 1 -interpolation and the theory of quasi-optimal meshes which are achievable approximations of the optimal meshes (Agouzal et al., 2009, 2010 [2] , [3] , Agouzal et al., 2010 [4] ).

We describe a new method for generating meshes that minimize the gradient of a discretization error. The key element of this method is construction of a tensor metric from edge-based error estimates. In our papers [1-4] we applied this metric for generating meshes that minimize the gradient of P"1-interpolation error and proved that for a mesh with N triangles, the L^2-norm of gradient of the interpolation error is proportional to N^-^1^/^2. In the present paper we recover the tensor metric using hierarchical a posteriori error estimates. Optimal reduction of the discretization error on a sequence of adaptive meshes will be illustrated numerically for boundary value problems ranging from a linear isotropic diffusion equation to a nonlinear transonic potential equation.

The paper is devoted to the analysis of optimal simplicial meshes which minimize the gradient error of the piecewise linear interpolation over all conformal simplicial meshes with a fixed number of cells NT. We present theoretical results on asymptotic dependencies of Lp-norms of the gradient error on NT for spaces of arbitrary dimension d. Our analysis is based on a geometric representation of the gradient error of linear interpolation on a simplex and a relaxed saturation assumption. We derive a metric field M p such that a M p-quasi-uniform mesh is quasi-optimal, for arbitrary d and p ∊ ]0, +∞]. Quasi-optimal meshes provide the same asymptotics of the Lp-norm of the gradient error as the optimal meshes.

We study an approximation for a class of non-Newtonian model, based on the general Ladyzhenskaya law for continuous and non-linear kinematic viscosity functions of the fluid. We introduce a non-conforming finite element approximation of non-linear Stokes problem; we give existence, uniqueness results and optimal error estimates under minimal regularity assumptions. Finally, we introduce a post-processing to obtain conforming approximation of stress, which allow to derive various equivalence with two or three field mixed dual approximations of non-linear Stokes problem.

In this Note, we show that a modified and simplified version of the estimator of Bank–Weiser can be used to define a robust a posteriori error estimator for singularly perturbed problem. We prove without comparison with a residual estimator or saturation assumption, the equivalence of the estimator with the error in the energy norm and the robusteness with respect to the diffusion coefficient. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

In this paper, we present an a posteriori error estimator for diffusion equation with the solution obtained by Black-box solver. The estimator is valid for classical primal, equilibrium and mixed finite element approximations with or without numerical integration.

Resume Dans cette Note, on montre qu'une version modifiee de l'estimateur de Bank–Weiser permet de definir un estimateur a posteriori en norme L∞ pour les methodes d'approximations conforme ou non conforme. On demontre, sans hypothese de saturation ni comparaison avec des estimateurs residuels, l'equivalence de l'estimateur avec la norme L∞ de l'erreur. Pour citer cet article : A. Agouzal, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 411–415.

Resume Dans cette Note on introduit une nouvelle formulation mixte a trois champs approchant les deplacements, le gradient des deplacements et les contraintes du probleme de l'elasticite non lineaire. Le probleme obtenu est non lineaire et sera approche par les espaces de type Raviart–Thomas. Nous montrons l'existence et l'unicite de la solution du probleme approche et etablissons des estimations a priori d'erreur optimales.

In this paper, first we discuss a technique to compare finite volume method and some well-known finite element methods, namely the dual mixed methods and nonconforming primal methods, for elliptic equations. These both equivalences are exploited to give us a posteriori error estimator for finite volume methods. This estimator is explicitly given, easy to compute and asymptotically exact without any regularity of the solution in unstructured grids.

In this paper we develop and analyse a new a posteriori error estimator for general Friedrichs' systems valid for most classical finite element approximations. This error estimator is based on comparison between an appropriate norm of the exact error, and the L 2 -norm of the residuals of the approximate solution. We prove that the estimator is independent of the dimension of the space and of the numerical approximation method used. Moreover the global majoration and local minoration constants are independent of the shape of the mesh.

A non-standard Equilibrium Method to solve some elliptic problems with non-standard boundary conditions is presented. A conforming primal variational formulation is used on the boundary, whereas an equilibrium one is established on the domain. One hereby obtains a coupled system, in which all matching conditions remain implicit. Existence and uniqueness results of the approximate solution are given, and the convergence rate of the method is specified for a model problem.

Resume On presente un estimateur d'erreur a posteriori pour le probleme de diffusion qui evalue l'erreur en loi de comportement : ∥▽ ũ h - p h ∥ 0,Ω , ou ũ h est une ≪ approximation ≫ conforme de la variable primale et p h est une approximation conforme de ▽ u obtenue par un post-traitement de l'approximation P 1 non conforme de u . Cette methode est etendue au probleme de Stokes. Des resultats similaires peuvent etre demontres pour le probleme de l'elasticite lineaire.