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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

We consider a data completion method for the Cauchy problem of Laplace equation, that is known as a highly ill-posed problem. For an approximate solution given by any one of algorithms of the data completion methods, we give the moment of the error between the true solution and the approximate one, which allows us to find an a posteriori explicit error estimation. This error indicator is used to improve the approximate solution, to give a knowledge of stability and can be used as a stopping criterion. Eventually, numerical experiments are provided.

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 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 derive a robust residual a posteriori error estimator for time-dependent convection–diffusion–reaction problem, stabilized by subgrid viscosity in space and discretized by Crank–Nicolson scheme in time. The estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time. Numerical experiments illustrate the theoretical performance of the error estimator.

The paper deals with the existence of solutions of some unilateral problems in the framework of Orlicz spaces.

In this paper, we present a new model of the European option price, it is about a model involving slow growth volatility. The SGV model (slow growth volatility model) is the name of the new model considered in this manuscript. The mathematical analysis shows that one has to resort to a degenerate parabolic equation, the resolution of which gives the price of the European option as a function of the time, the price of the underlying asset and the instantaneous volatility.

We derive a residual a posteriori error estimates for the subscales stabilization of con- vection diffusion equation. The estimator yields upper bound on the error which is global and lower bound that is local.

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

We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Peclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Peclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.

In this work, we study the error in the approximation of the solution of elliptic partial differential equations obtained with the nonconforming finite elements method; we adopt the error in a constitutive law approach.

We derive a residual a posteriori error estimator for the algebraic orthogonal subscales stabilization of convective dispersive transport equation. The estimator yields upper bound on the error which is global and lower bound that is local. Numerical studies show the behaviour of the error indicator and how it is robust to deal with singularities. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We give some diffusion models which can be used for restoration in image processing. The proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. The existence theorem for solutions and some numerical results are given.

This paper suggests a decoupled method in order to approximate a free boundary separating the depletion region and the charge neutrality region in a field effect transistor of MESFET type. In order to do that, a simplified drift-diffusion model is used. A decoupled algorithm is presented in (Math. Comput. Simulat. 1998; 47(6):531–539). In this work an existence theorem is proved, and the study of the convergence of the previous algorithm is presented with some numerical results. Copyright © 2005 John Wiley & Sons, Ltd.

The constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is the framework of finite element approximations of systems of partial differential equations. We consider an approximation of general systems of linear partial differential equations in R 3. Concerning a multilevel convergence rate corresponding to nested general tetrahedral meshes of size h and 2h, we give an estimate of this constant for general three-dimensional cases.

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three-dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large-scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real-life elasticity problems. Copyright © 2001 John Wiley & Sons, Ltd.

Resume Dans cette Note, nous montrons l'existence et l'unicite de la solution d'un probleme de frontiere libre formule en un probleme d'optimisation de forme. La demonstration de ce resultat est basee sur le principe de Schauder [6].

We prove an existence result for solutions of nonlinear parabolic inequalities with data in Orlicz spaces.

The paper deals with the existence of solutions of some parabolic bilateral problems approximated by the renormalized solutions of some parabolic equations.

Pour etudier le probleme du diffuseur d'une turbine hydraulique, nous avons utilise le lagrangien pour etablir formellement les equations adjointes et faire un calcul rapide du gradient de la fonctionnelle. En utilisant la derivee particulaire, nous avons calcule ce gradient permettant d'effectuer un deplacement de la frontiere sans perte de regularite sur le domaine. L'application d'une variante du theoreme de Min-Max permet de se passer du theoreme des fonctions implicites. Des resultats numeriques etayant la validite du modele sont presentes.