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This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity problem. It is shown that this family passes the patch-test, and that by suitably adding some orthogonality conditions, on a general class of enrichment functions, we can derive higher order consistency error estimates. Our general theoretical results, see Theorems 4.1 and 4.2, unify, simplify and extend a number of existing works on the improvement of the order of consistency error. Numerical experiments are carried out to demonstrate that our method is optimal for various Lame parameter μ, shear modulus λ and locking free when the Poisson parameter ν approaches close to 0.5.

A numerical method for simulating transient flows of gas-liquid mixtures is proposed. The mathematical model, established for a suspension of gas bubbles in liquid, includes an equation taking into account the relative velocity between the gas and liquid. A numerical technique based on the Mac Cormack scheme combined with the method of characteristics is presented. Theoretical results for transients initiated by a rapid closing valves are compared with measurements. A good agreement is found particularly for large values of initial dissolved gas concentration.

In this article we study a projection-stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery-based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017

In this paper, we will study the fully discretized finite element approximation for an incompressible flow in porous media. The model consists of the heat equation, the equation for the concentration and the equations of motion under the Darcy law. The model is rewritten using the stream function-vorticity formulation. The Stability of the fully discrete problem is established. Optimal a priori error estimates are given.

We develop in this article an a posteriori error estimator for the P1-nonconforming finite element approximation, for a diffusion-reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015

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.

In this paper, we derive an a posteriori error estimator, for nonconforming finite element approxi- mation of convection-diffusion equation. The a posteriori error estimator is based on the local problems on stars. Finally, we prove the reliability and the efficiency of the estimator without saturation assumption nor comparison with residual estimator

In this paper we suggest a fully discretized problem of a model describing two-step chemical ki- netics. The model considered is a system of equations coupling Navier-Stokes equations with three non-linear reaction-diffusion equations. A space-time finite elements approximations are presented. The stability of the fully discretized problem is studied. Optimal error estimates are given.

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection-diffusion equations with height Peclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 1717–1728, 2012

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming P1 space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in W 1,p(Ω \ Γ ) ∩ W 2,s(Ω \ Γ ), the interpolant Ihu defined by this new space satisfies

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.

University Lyon1, Institute Camille Jordan, UMR 5208, 69100 Villeurbanne, FranceAbstract. In this paper, a new a posteriori error estimator for nonconforming convection diffu-sion approximation problem, which relies on the small discrete problems solution in stars, hasbeen established. It is equivalent to the energy error up to data oscillation without any saturationassumption nor comparison with residual estimator.Key words: a posteriori error estimator, nonconforming nite elements method, convection diffu-sion equationsAMS subject classication: 65N30, 65N50

International audience; The paper deals with the existence and uniqueness of solutions of some non linear parabolic inequalities in the Orlicz-Sobolev spaces framework.

A conservative method to solve waste repository problems with non-standard interface conditions is presented. One hereby obtains a coupled system, in which all matching conditions remain implicit. A numerical scheme is proposed and stability analysis as well as error estimates are provided.

A primal hybrid finite element scheme is introduced to produce completely discontinuous solution for diffusion and convection-diffusion problems. Same rate of convergence as classical methods is obtained in suitable norms. Finally an a posteriori error estimator is given.

The paper is devoted to the stability of reaction fronts in thin domains. The influence of natural convection and of heat losses through the walls of the reactor is studied numerically and analytically. Critical conditions of stability of stationary solutions are obtained.

The saturation assumption is widely used in a posteriori error analysis of finite element methods. It asserts, in its simplest form, that the solution can be approximated asymptotically better with quadratic than with linear finite elements. In this article, we show that a simple modification of this “hypothesis” is valid, and the proof of many authors can be made rigorous with this simple modification. We prove also the robustness of the estimator for a singularly perturbed problem. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004

SUMMARY This paper deals with numerical analysis of system coupling Navier-Stokes equations with two non- linear reaction-diusion equations. This system modelize a propagation of reaction front at the case of polymerization. Anite elements approximation is presented, the existence and uniqueness are estab- lished. Optimal error estimates are given. Copyright ? 2003 John Wiley & Sons, Ltd.

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.

International audience; Lotkaian informetrics is the framework most often used to study statistical distributions in the production and usage of information. Although Lotkaian distributions are traditionally used to characterize the Information Production Process (IPP) , we have shown in a previous article that the IPP can successfully be studied using the effort function – the latter having been initially introduced to define the Exponential Informetric Process (EIP). These themes continue to be developed in this article , in which we present a necessary and sufficient condition for the existence of the EIP. Our current approach is similar to the one used to study IPPs. Inverse power and exponential distributions serve to illustrate the results obtained in the context of an EIP. Numerical examples are discussed .

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.

We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed nite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.

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.

This paper deals with the solution of problems involving partial differential equations in $\mathbb{R}^3$ . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented which uses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitable norms.