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Residual error estimators for the timedependent Stokes equationsN. Kharrat * — Z. Mghazli *** LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, Faculté des Sciences de SfaxB.P.37, 1002 Tunis-BelvédèreTUNISIEnizar.kharrat@enit.rnu.tn** Laboratoire SIANOUniversité Ibn Tofaïl, Faculté des SciencesB.P. 133, 14000 KénitraMarocmghazli_zoubida@yahoo.comABSTRACT. We present a posteriori residual error estimators for the approximate time-dependentStokes model Chorin-Temam [4] projection scheme using a conforming finite element discretization.We prove a time-space global upper bound and local space lower bounds for the error.RÉSUMÉ. Nous présentons des estimateurs d’erreurs a posteriori par résidus pour le modèle deStokes instationnaire approché par le schéma de projection de Chorin-Temam [4] en utilisant unediscrétisation par éléments finis conformes. Nous prouvons une majoration globale de l’erreur spatiotemporelle,ainsi que des minorations locales sur l’erreur spatiale.KEYWORDS : A posteriori estimators, residual error, time-dependent Stokes model, projection scheme,global upper bound, local lower bound.MOTS-CLÉS : Estimateurs a posteriori, erreurs par résidu, modèle de Stokes instationnaire, schémade projection, majoration globale, minoration locale.TAMTAM –Tunis– 2005 202
REE for Stokes equations 2031. Introduction and preliminariesLet Ω be a bounded connected domain of R d (d = 2, 3), with Lipschitz-continuousboundary Γ and we denote by ⃗n the unit outward normed vector to Ω on Γ. Let T > 0 bea real constant. We set X = H0 1 (Ω) d , Y = L 2 (Ω) d and M = H 1 (Ω)/R.For an initial velocity field, u 0 ∈ H = {v ∈ Y ; ∇.v = 0 in Ω , v.⃗n = 0 on Γ }, dataf ∈ L 2 (0, T ; H), and a given real kinematic viscosity ν > 0, we consider the incompressibletime-dependent Stokes problem :Find (u(t), p(t)) ∈ X × L 2 (Ω)/R such that for almost every t ∈]0, T [⎧d ⎪⎨ (u, v) + ν(∇u, ∇v) − (p, ∇.v) = (f, v) ∀ v ∈ X,dt(q, ∇.u) = 0 ∀ q ∈ L ⎪⎩2 (Ω)/R ,(1)u(0) = u 0 .Let us introduce a regular partition 0 = t 0 < t 1 < ... < t N = T of the time intervalτ n−1[0 , T ], with step sizes τ n = t n − t n−1 such that the regularity parameter max is1≤n≤N τ nbounded from above independently of τ. We denote by τ the N−tuplets (τ 1 , ..., τ N ) andby f n = 1 ∫ tnf(t)dt for each 1 ≤ n ≤ N. The variational form of the semi-discreteτ n t n−1Chorin-Temam projection scheme applied to the Stokes equations (1), reads :Find (ũ n , u n , Φ n ) 1≤n≤N ∈ (X × Y × M) N initialized by u 0 = u 0 and satisfying(ũ n , v) + ντ n (∇ũ n , ∇v) = (u n−1 , v) + τ n (f n , v) ∀v ∈ X, (2){ (u n , v) + τ n (∇Φ n , v) = (ũ n , v) ∀ v ∈ Y,(u n (3), ∇q) = 0 ∀ q ∈ M.By standard arguments, it is readily checked that problems (2) and (3) are well posed.Indeed step (2) constitutes an elliptic boundary value problem for an intermediate velocityunknown ũ n , which is a prediction of u(t n ) satisfying a homogeneous Dirichlet boundarycondition, but is not divergence free. The second step (3) represents a Darcy’s problemwhich determines the end-of-step divergence-free velocity u n which is a correction ofũ n , together with a suitable approximation of the pressure distribution Φ n . We easilycheck that Φ n is the solution of a Poisson problem with homogeneous Neumann boundaryconditions. We assume thar Ω is polygonal or polyhedral (d = 2 or 3). For each n,0 ≤ n ≤ N τ , we associate a regular triangulation Thn of Ω into triangles or tetrahedra. Foreach element K in Th n,we denote by h K the diameter of K, E K the set of edges or facesE of K wich are not contained in Γ and for each element E in E K we denote by h E thediameter of E. The fully discrete version of algorithm (2)-(3) in the framework of spatialGalerkin finite element approximation takes similar formulation written for the unknowsequencedenoted by (ũ n h , un h , Φn h ) 1≤n≤N τretrieved in some appropriate approximationsubspaces of X, Y and M.TAMTAM –Tunis– 2005
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PréfaceOrganiser la seconde éditi
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Comité d’organisationMohamed Abd
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SOMMAIRE / CONTENTSI Conférences I
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Equilibre dans un système dynamiqu
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Arlequin method: Practical impacts
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Application de la méthode de Gauss
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Conférences InvitéesInvited Prese
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4 Amara et al.1. IntroductionWe are
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6 Amara et al.2.2. The nonlinear Na
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8 Amara et al.4. Numerical resultsW
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10 Auger et al.1. Introduction :Dan
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12 Auger et al.La condition de stab
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14 Auger et al.3.1. Taux de migrati
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16 Auger et al.L’équilibre rapid
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18 Auger et al.[7] BRAVO DE LA PARR
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Quelques aspects mathématiques etn
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22 El Alaoui Talibi M.- Modèle d
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24 El Alaoui Talibi M.[3] M. Chamba
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26 Habbal1. IntroductionAngiogenesi
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28 HabbalThe pressure drop denoted
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30 Habbaltheorem 1 There exists a N
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32 HabbalFigure 2. Second Nash loop
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34 Hauray et al.1. IntroductionWe a
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36 Hauray et al.minimal time interv
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38 Hauray et al.From this last theo
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Approximation de l’opérateur d
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42 LemrabetLorsque δ → 0, l’é
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44 Lemrabetavec[∆ ∗ (δ)c = [I
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46 Lemrabet6. Bibliographie[1] N. B
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48 Jaafar-Belaid et al.1. Introduct
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50 Jaafar-Belaid et al.where f(ρ)
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52 Jaafar-Belaid et al.202040406060
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IIContrôleControl55
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58 Arfaoui1. IntroductionCe travail
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60 ArfaouiLe système adjoint assoc
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62 Arfaoui[3] H. ARFAOUI , « Comma
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64 Benzekri et al.1. Global control
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66 Benzekri et al.ẏ = ∂H∂x =
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0 1 2 3 4 5 668 Benzekri et al.32.5
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70 Akian et al.1. IntroductionWe co
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72 Akian et al.We prove that v t+δ
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.74 Akian et al.0.0−1.0Exact solu
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76 Bokanowski et al.1. General fram
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78 Bokanowski et al.One interesting
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80 Bokanowski et al.4. Numerical ex
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Sur les problèmes de contrôle opt
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84 Metouileur lorsque u est dans L
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86 MetouiThéorème 2 Soit ū ε le
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88 Metoui[5] L.S. HOU, S.S. RAVINDR
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Système d’information intégré
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Gestion et modélisation des ressou
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Random perturbations of reduced gra
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Global Optimization of Water Resour
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Global Optimization of Water Resour
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IVEquations DifférentiellesDiffere
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114 B. Abdelkader1. IntroductionTel
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116 B. AbdelkaderNotons que les con
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118 B. AbdelkaderAlors pour tout X
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120 Bouzahir1. IntroductionWe consi
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122 Bouzahirwhere the linear operat
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Goursat boundary value problem forh
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126 A. Guezane-Lakoud et al.3. Func
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128 A. Guezane-Lakoud et al.5. Exis
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ACP neuronale : application à l’
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132 Chitroubl’approximation de la
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134 Chitroubcompression égal à 3.
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136 Chitroub[4] J. P. Nougier, Mét
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138 Ferchichi et al.1. Introduction
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140 Ferchichi et al.soit T −1 = (
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142 Ferchichi et al.[4] G.I. BISCHI
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144 Hafidi1. IntroductionOn étudie
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146 Hafidia ) ∫ Ωpdx → +∞ qu
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148 HafidiLEMME 3.2. On a lim R(θ
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150 Lefebvre1. IntroductionSoit (X
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Page 174 and 175: 156 Meskine1. IntroductionDans le p
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Page 178 and 179: 160 Meskine[12] E. YA. KHRUSLOV, L.
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Page 188 and 189: 170 K. SaoudiRemarque 3.1. Du lemme
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Page 196 and 197: 178 Achchab et al.1. IntroductionDu
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Page 200 and 201: 182 Achchab et al.[2] B.ACHCHAB, M.
Page 202 and 203: 184 Alla et al.1. IntroductionLes e
Page 204 and 205: 186 Alla et al.3. Position du probl
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Page 210 and 211: 192 Achchab et al.où T hi/2 est le
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Page 214 and 215: 196 El Dabaghi et al.1. Motivation
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Page 227: VIMécanique des FluidesFluid Mecha
Page 230 and 231: 212 Abdelwahed et al.1. Introductio
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Page 234 and 235: 216 Abdelwahed et al.Figure 3. Isov
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Page 250 and 251: 232 Amoura et al.1. IntroductionFor
Page 252 and 253: 234 Amoura et al.where u = rot r ψ
Page 254 and 255: 236 Amoura et al.CURRENT FUNCTION5
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Page 258 and 259: 240 Frih et al.et dans la fracture,
Page 260 and 261: 242 Frih et al.6. Bibliographie[1]
Page 262 and 263: 244 Hasnaoui et al.1. IntroductionL
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Page 266 and 267: 248 Hasnaoui et al.5. Conclusion et
Page 268 and 269: 250 Hernane-Boukari1. IntroductionW
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252 Hernane-Boukarianda 3 =−11 +
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254 Kloucha et al.1. Motivation et
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256 Kloucha et al.Si le domaine Λ
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INRIA-MODULEF0INRIA-MODULEF.250INRI
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Dimensions fractales : attracteurs
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262 Lamrini Uahabi et al.4. Attract
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264 Lamrini Uahabi et al.l 0 = λ
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266 Lamrini Uahabi et al.En conséq
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Numerical analysis for the problem
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270 El-Kyal et al.d(u) = 1 2(∇u +
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272 El-Kyal et al.Theorem 3.1 There
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274 Marchand et al.1. IntroductionD
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276 Marchand et al.On remarque en p
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278 Marchand et al.dans la formulat
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280 Mezali et al.1. Motivation et m
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282 Mezali et al.s’annulant sur c
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284 Mezali et al.Temps CPUTemps Ela
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R. Touihri & S. Ghnimi *Méthode de
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288 Touihri et al.Les équations go
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290 Touihri et al.Tol N It (Newt) t
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292 Touihri et al.5. Bibliographie[
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Décomposition de domaine pour un m
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DDM pour les fractures 297et−→
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DDM pour les fractures 299dont l’
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Arlequin method: Practical impacts
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Arlequin method 303of the Arlequin
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Arlequin method 3052.2. Penalized A
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Arlequin method 307refinements of t
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Reduction methods and uncertaintypr
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Reduction methods and uncertainty p
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Reduction methods and uncertainty p
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0.00e+00 1.00e−11 2.00e−11 3.00
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Homogénéisation d’un problème
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Problème de conduction-rayonnement
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Problème de conduction-rayonnement
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Equation d’Hamilton-Jacobi 3231.
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Equation d’Hamilton-Jacobi 325qui
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0Equation d’Hamilton-Jacobi 3272
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Détermination du nombre de région
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Simulation des courants de Foucault
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Une méthode d’accélération de
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Schéma SRNHS 3491. IntroductionUn
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Schéma SRNHS 3513. Application du
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Schéma SRNHS 3535. ConclusionLes r
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Smoothing and compressing surfaces
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Problème de convection en milieu p
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Condensation de masse pour la méth
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Condensation de masse 367montre que
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Condensation de masse 369Le terme h
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Condensation de masse 371Aet Qile f
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VIIIModèles CinétiquesKinetic Mod
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376 Ben Abdallah et al.1. Introduct
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378 Ben Abdallah et al.where B ∓
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380 Ben Abdallah et al.ture that th
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382 Degond et al.1. IntroductionOn
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384 Degond et al.q φ s (x, t)/(|q|
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386 Degond et al.Enfin l’opérate
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Les métaheuristiques : application
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Les métaheuristiques 391simulé fu
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Les métaheuristiques 393Fig. 1 Dia
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Les métaheuristiques 395Fig.4 : R
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Prices after capacity addition in m
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Multi-user elastic demand communica
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Shape optimization 4051. Introducti
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Shape optimization 4072.1. Variatio
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Shape optimization 409(a)(b)(c)(d)F
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Problèmes d’optimisation en éco
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Assimilation de données pour l’e
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Écart à la Réciprocité et ident
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Identification de fissures planes 4
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Algorithme de Kohn et Vogelius 4331
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Algorithme de Kohn et Vogelius 435l
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Algorithme de Kohn et Vogelius 4375
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Résolution d’un problème de Cau
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Problème inverse géométrique 445
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Identification de fissures 4511. In
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−0.5 −0.4 −0.3 −0.2 −0.1
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Application de l’algorithme deNeu
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Complétion de données en élastic
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Complétion de données en élastic
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Analytic extensions on an annulus:a
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Analytic extensions on an annulus 4
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Analytic extensions on an annulus 4
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−1 −0.8 −0.6 −0.4 −0.2 0
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Cauchy problem for Laplace’s equa
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Modélisation asymptotique des onde
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Ondes de relief 479 uuuv 1 1 pu v
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Ondes de relief 481( )ˆeiZw( C e D
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Ondes de relief 4836. Bibliographie
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Régularisation pour l’aéroacous
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Miroir à retournement temporel 491
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Vibro-acoustique instationnaire 497
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Vibro-acoustique instationnaire 499
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Ondes dans les milieux poroélastiq
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The implicitly restarted Arnoldi me
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¨§¨ §IRAM method 509To overcome
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IRAM method 511The application of t
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516 Achchab1. IntroductionDans la m
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518 AchchabNous montrons dans le pa
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520 Achchab5. Bibliographie[1 ] AIE
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522 Derbel et al.1. IntroductionLa
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524 Derbel et al.On se propose de d
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A Stochastic partial differential e
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528 El Saadidiffusion process to a
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530 El Saadiwhere δ Xi(t) is the D
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532 El Saadi5. References[1] A. L.
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534 Ben Miled et al.1. Introduction
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536 Ben Miled et al.nature de l’e
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538 Ben Miled et al.Ensuite on étu
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Optimal spatial distribution of a b
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542 Mchich et al.at time t. The com
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544 Mchich et al.3∑where r = r i
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XIIIStructureStructure547
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550 Ayadi1. IntroductionConsider a
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552 AyadiFor all 0 ≤ p ≤ i, P p
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554 AyadiFigure 3. The curves above
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Analyse mathématique et numérique
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558 Chamekh et al.2. Modèle de tig
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560 Chamekh et al.Γ sr(s)r(σ)Figu
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Modélisation de l’effet dynamiqu
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564 RahmaniI 2 étant l’identité
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566 Rahmani• ( u 1δ1∫+ ,0u 1δ
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Résolution d’un systéme d’ela
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570 Saidoù u vérifie :n∑n∑j=1
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572 SaidEn effet :∣ u(y, s) 1+α
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Régularisation d’un problème d
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576 Achchab et al.On définit : F +
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578 Achchab et al.3. Estimation a p
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Modélisation d’un pieuR. Aboulai
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582 Aboulaich et al.Figure 1. Struc
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584 Aboulaich et al.5. Résultats n
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XIVThéorie des GraphesGraph Theory
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590 Hlaoui1. IntroductionMore and m
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592 Hlaoui3. The New Graph Represen
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594 Hlaoui1 1 2 3 4 1 1 3 411 421 4
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596 M. Nekri et al.1. IntroductionL
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598 M. Nekri et al.proposés. Ces m
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600 M. Nekri et al.cycle de longueu
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