F. Ben Belgacem

Hybrid finite element methods for the Signorini problem

We study three mixed linear finite element methods for the numerical simulation of the two-dimensional Signorini problem. Applying Falks Lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them. Two of these finite elements provide optimal results under reasonable regularity assumptions on the Signorini solution, and the numerical investigation shows that the third method also provides optimal accuracy.

Numerical Simulation of Some Variational Inequalities Arisen from Unilateral Contact Problems by the Finite Element Methods

This contribution deals with the finite element approximation of the variational inequalities coming from the Signorini problem and the unilateral contact between two elastic bodies. The numerical analysis we carry out improves former results and states optimal convergence rates, under reasonable regularity hypotheses, for both conforming and nonconforming methods.

The Mortar Finite Element Method for 3D Maxwell Equations: First Results

In the framework of domain decomposition methods, we extend the main ideas of the mortar element method to the numerical solution of Maxwells equations (in wave form) by H( curl)-conforming finite elements. The method we propose turns out to be a new nonconforming, nonoverlapping domain decomposition method where nonmatching grids are allowed at the interfaces between subdomains. A model problem is considered, the convergence of the discrete approximation is analyzed, and an error estimate is provided. The method is proven to be slightly suboptimal with a loss of a factor

Inf-sup conditions for the mortar spectral element discretization of the Stokes problem

\scriptstyle\sqrt{|{\rm ln} h|}

Quadratic finite element approximation of the Signorini problem

with respect to the degree of polynomials. In order to achieve this convergence result we nevertheless need extra-regularity assumptions on the solution of the continuous problem.

The hp-mortar finite-element method for the mixed elasticity and stokes problems

Summary. The aim of this paper is to prove some Babuška–Brezzi type conditions which are involved in the mortar spectral element discretization of the Stokes problem, for several cases of nonconforming domain decompositions.Résumé. Le but de cet article est de prouver la condition inf-sup de Babuška–Brezzi qui intervient dans la discrétisation par éléments spectraux du problème de Stokes, dans le cas de plusieurs décompositions de domaine non conformes traitées par méthode de joints.

The Lavrentiev regularization of the data completion problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved With the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem. responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falks Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

Numerical simulation of the wave equation with discontinuous coefficients by nonconforming finite elements

Abstract The motivation of this work is to apply the hp -version of the mortar finite-element method to the nearly incompressible elasticity model formulated as a mixed displacement-pressure problem as well as to Stokes equations in primal velocity-pressure variables. Within each subdomain, the local approximation is designed using div-stable hp -mixed finite elements. The displacement is computed in a mortared space, while the pressure is not subjected to any constraints across the interfaces. By a Boland-Nicolaides argument, we prove that the discrete saddle-point problem satisfies a Babuska-Brezzi inf-sup condition. The inf-sup constant is optimal in the sense that it depends only on the local (to the subdomains) characteristics of the mixed finite elements and, in particular, it does not increase with the total number of the subdomains. The consequences, that we are aware of, of such an important result are twofold. 1. • The numerical analysis of the approximability properties of the hp-mortar discretization for the mixed elasticity problem allows us to derive an asymptotic rate of convergence that is optimal up to “ log ” p in the displacement; this is addressed in the present contribution. 2. • When the mortar discrete problem is inverted by substructured iterative methods based on Krylov subspaces with block preconditioners, in view of the results for conforming finite elements [1], the condition number of the solver should grow logarithmically on ( p , h ) and not depend on the total number of the subdomains.

Identifiability of surface sources from Cauchy data

The Lavrentiev regularization method is naturally fitted to the data completion problem, put under the variational form proposed by Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915). We address the important issue of selecting the regularizing parameter. We study an a priori choice and the a posteriori choice by means of the discrepancy principle written on the Kohn?Vogelius function. In both cases, we prove error estimates similar to those expected for Tikhonovs method.

An ill-posed parabolic evolution system for dispersive deoxygenation?reaeration in water

The goal of this article is to apply the mortar finite element method to the numerical simulation of (elec- tromagnetic and/or acoustic) waves propagating in an inhomogeneous support. This approach allows us to use meshes well adapted to the local physical parameters of the media without any conformity constraints. A complete mathematical study is supplied providing the expected optimal convergence rate. Numerical performances of such a technique, as well as its advantages, are also discussed.