In a previous article about the homogenization of the classical problem of
diff usion in a bounded domain with su ciently smooth boundary we proved that
the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently
smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions
we show that in any open set strongly included in the error is of order
$\epsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or
polyhedral (n=3) boundary we also give the global and interrior error
estimates

Bensoussan A.

Cioranescu D.

GEORGES GRISO

Griso G.

Grisvard P.

Oleinik O. A.

English

arXiv

Griso, Georges

Numérisation de Documents Anciens Mathématiques

1/eriodicureestimationstionritédansnceC. R. Acad. Sci. Paris, Ser. I 340 (2005) 251–254http://france.elsevier.com/direct/CRASSCalculus of Variations/Partial Differential EquationsInterior error estimate for periodic homogenizationGeorges GrisoUniversité Pierre et Marie Curie (Paris VI), laboratoire Jacques-Louis Lions (analyse numérique) 4, place Jussieu,75252 Paris cedex 05, FranceReceived 13 October 2004; accepted 20 October 2004Available online 4 January 2005Presented by Philippe G. CiarletAbstractThe aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the punfolding method. The interior error estimates areobtained by transposition without any supplementary hypothesis of regularityon correctors. This error is of orderε. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméEstimation intérieure de l’erreur en homogénéisation périodique. On s’intéresse dans cette Note à l’erreur intériedans les problèmes d’homogénéisation périodique. Les techniques utilisées sont celles de l’éclatement périodique. L’intérieure de l’erreur est obtenue par lméthode de transposition sanfaire d’hypothèses supplémentaires de régularité sur lecorrecteurs. Cette erreur est d’ordreε. Pour citer cet article : G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeCette Note présente la suite d’un travail [4,5] sur l’estimation de l’erreur dans le problème d’homogénéisapériodique. Dans ces articles on a montré que l’erreur est d’ordreε1/2 sous des hypothèses minimales de régulades correcteurs. On montre ici que l’erreur intérieure est d’ordreε.Les premières estimations de l’erreur dans le problème d’homogénéisation périodique ont été données[1,3,6]. Dans ces travaux on fait l’hypothèse que les correcteurs appartiennent àW1,∞(Y ) (Y = ]0,1[n est lacellule de référence) ; l’estimation obtenue était enε1/2.La première partie de cette Note est consacréeà d s théorèmes de projection. Le domaineΩ st de frontièrelipschitzienne. On donne (Théorème 2.1) un majorant de la distance entre l’éclaté d’une fonction deH 1(Ω) etl’espaceH 1per(Y ;L2(Ω)) pour la norme deH 1(Y ; (H 1(Ω))′) puis (Théorème 2.2) un majorant de la distaentre l’éclaté du gradient d’une fonction deH 1(Ω) et l’espace∇xH 1(Ω) ⊕ ∇yH 1per(Y ;L2(Ω)) pour la normeE-mail address:georges.griso@wanadoo.fr (G. Griso).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.10.027252 G. Griso / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 251–254nesurer lau caslettentr le. Laitionularitynttedthefeicehetheedyde [L2(Y ; (H 1(Ω))′)]n. Ces résultats s’appuient sur le théorème mesurant le défaut de périodicité d’une fonctioharmonique appartenant àH 1(Y ) (voir [5]). Les majorants obtenus dépendent à la fois deε, de la norme de lafonction dansH 1(Ω) et de la normeH 1 de cette fonction dans un voisinage d’ordreε de la frontière deΩ .La seconde partie de cette étude est consacrée à l’estimation intérieure de l’erreur. On cherche à mdistance entre la solution d’un problème de diffusion à coefficients périodiques et fortement oscillants(3) et alimite qui est la solution du problème(5) donné sous la forme éclatée. Dans cette Note on se restreint ad’un domaineΩ ⊂ Rn de frontière assez régulière (C1,1), de la condition aux limites homogène de Dirichet d’un second membre appartenant àL2(Ω). Dans une première étape on part du problème(3) en choisissancomme fonction-testΨ + ε ∑ni=1 ρεQε( ∂Ψ∂xi )χi( .ε ), Ψ ∈ H 10 (Ω) ∩ H 2(Ω). On transforme ensuite par éclateml’égalité obtenue puis on applique le Théorème 2.2 à la fonctionϕε. L’estimation supplémentaire demandée paThéorème 2.2 du gradient deϕε au voisinage de la frontière deΩ est obtenue grâce au Théorème 4.1 de [5]méthode de transposition donne l’estimationL2 de l’erreur. Dans une seconde étape on obtient l’estimationH 1 del’erreur dans tout ouvert fortement inclu dansΩ grâce à l’estimationL2 précédente. Ces erreurs sont d’ordreε.Dans un article à venir on donnera les démonstrations détaillées et on ajoutera à cette étude le cas de la condaux limites homogène de Neumann et le cas d’un ouvert de frontière polygonale (n = 2) ou polyédrale (n = 3).Ce travail utilise les notations de [2,4,5].1. IntroductionIn the periodic homogenization problem we have obtained an error estimate (in [5]) without any reghypothesis on the correctors. It was supposed that the solution of the homogenized problem belongs toH 2(Ω).This holds true with a smooth boundary and homogeneousDirichlet or Neumann limit conditions. The exponeof ε in the error estimate is equal to 1/2.The aim of this work is to give further error estimates with again minimal hypotheses on the correctors.Section 2 is dedicated to Theorem 2.2 which is the essential tool to obtain new estimates. This theorem is relato the periodic unfolding method (see [2,5]). We show that for anyϕ in H 1(Ω), whereΩ is a bounded open seof Rn with Lipschitz boundary, there exists a functionϕ̂ε in L2(Ω;H 1per(Y )), such that the distance between tunfoldedTε(∇xϕ) and ∇xϕ + ∇y ϕ̂ε is of orderε in the space[L2(Y ; (H 1(Ω))′)]n provided that the norm ogradientϕ in a neighbourhood (of orderε) of the boundary ofΩ is less thanε1/2. This result is based on ththeorem that measures the periodic defect of a harmonic function belonging toH 1(Y ) (see [5]).In Section 3, we give the distance between the solution of a diffusion problem with strongly oscillating periodcoefficients(3) and its limit. This limit is the solution of problem(5) given in its unfolded form. We consider thcase of an open setΩ with sufficiently smooth boundary (C1,1) and homogeneous Dirichlet limits conditions, tright-hand side of the homogenized problem being inL2(Ω). In Theorem 3.1, we show by transposition thatL2 error estimate and the interior error estimate is of orderε. The required condition in Theorem 2.2 is obtainthanks to the estimates of Theorem 4.1 of [5].The detailed demonstrations including the case of a domainΩ with polygonal (n = 2) or polyhedral boundar(n = 3) will be presented in forthcoming article.Throughout this study we use the notation of [2,4,5].2. Preliminary resultLet Ω be a bounded domain inRn with a Lipschitzian boundary. We denoteΩ̂ε,k ={x ∈ Ω | dist(x, ∂Ω) < k√nε}, Ω̃ε,k = {x ∈ Rn | dist(x,Ω) < k√nε}, k ∈ {1,2,3},Ωε = Int( ⋃ξ∈Ξεε(ξ + 	Y )), Ξε ={ξ ∈ Zn | ε(ξ + Y ) ∩ Ω = ∅}, Y = ]0,1[n.There exists a linear and continuous extension operatorP from H 1(Ω) into H 1(Ω̃ε,3) such thatG. Griso / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 251–254 253he:c-te‖∇P(ϕ)‖[L2(Ω̃ε,3\Ω)]n  C‖∇ϕ‖[L2(Ω̂ε,3)]n and ‖P(ϕ)‖L2(Ω̃ε,3)  C(‖ϕ‖L2(Ω) + ε‖∇ϕ‖[L2(Ω̂ε,3)]n).In the sequel any function belonging toH 1(Ω) is extended into a function belonging toH 1(Ω̃ε,3). The extensionof ϕ is still denotedϕ. We now can define and evaluate the local averageMεY (ϕ) (respectively theQ1-interpolateQε(ϕ)) in all cells contained iñΩε,3 (resp.Ω̃ε,2) (see [2] for details).Theorem 2.1. For anyϕ belonging toH 1(Ω), there existsψ̂ε in L2(Ω;H 1per(Y )) such that‖ψ̂ε‖L2(Ω;H1(Y ))  C{‖ϕ‖L2(Ω) + ε‖∇ϕ‖[L2(Ω)]n},‖Tε(ϕ) − ψ̂ε‖H1(Y ;(H1(Ω))′) Cε{‖ϕ‖L2(Ω) + ε‖∇ϕ‖[L2(Ω)]n} + C√ε{‖ϕ‖L2(Ω̂ε,2) + ε‖∇ϕ‖[L2(Ω̂ε,2)]n}.(1)The constants depend only onΩ .In [5] we proved that‖Tε(ϕ) − ψ̂ε‖H1(Y ;H−1(Ω)))  Cε{‖ϕ‖L2(Ω) + ε‖∇ϕ‖[L2(Ω)]n} since for any functionΨ ∈ H 10 (Ω), we have‖Ψ ‖L2(Ω̂ε,2)  Cε‖∇Ψ ‖[L2(Ω)]n .If Ψ belongs toH 1(Ω), we only have‖Ψ ‖L2(Ω̂ε,2)  C√ε‖Ψ ‖H1(Ω). This accounts for the presence of tsupplementary termC√ε{‖ϕ‖L2(Ω̂ε,2) + ε‖∇ϕ‖[L2(Ω̂ε,2)]n} in the second estimate of(1). The proof is similar tothat of Proposition 3.3 of [5]. We measure the periodic defect (cf. [4,5]) of the unfolded ofϕ, Tε(ϕ) in the spaceH 1(Y ; (H 1(Ω))′), then apply Theorem 2.3 of [5].Theorem 2.2. For anyϕ belonging toH 1(Ω), there existŝϕε in L2(Ω;H 1per(Y )) such that{‖ϕ̂ε‖L2(Ω;H1(Y ))  C‖∇ϕ‖[L2(Ω)]n,‖Tε(∇xϕ) − ∇xϕ − ∇yϕ̂ε‖[L2(Y ;(H1(Ω))′)]n  Cε‖∇ϕ‖[L2(Ω)]n + C√ε‖∇ϕ‖[L2(Ω̂ε,3)]n .(2)The constants depend onlyΩ .The idea is to takeϕ in H 1(Ω) and write it asϕ = Φ + εϕ̃ whereΦ = Qε(ϕ). We use the following estimate‖∇Φ‖[L2(Ω)]n + ‖ϕ̃‖L2(Ω) + ε‖∇ϕ̃‖[L2(Ω)]n  C‖∇ϕ‖[L2(Ω)]n (see [2]), and apply Theorem 2.1 to the funtion ϕ̃. We obtain‖ϕ̂ε‖L2(Ω;H1(Y ))  C‖∇ϕ‖[L2(Ω)]n and‖Tε(ε∇xϕ̃) − ∇y ϕ̂ε‖L2(Y ;(H1(Ω))′)  Cε‖∇ϕ‖[L2(Ω)]n +C√ε‖∇ϕ‖[L2(Ω̂ε,3)]n (recall thatTε(ε∇x ϕ̃) = ∇yTε(ϕ̃)).As in Theorem 3.4 of [2] we prove that∥∥Tε(∇xΦ) − ∇xΦ∥∥[L2(Y ;(H1(Ω)′)]n  Cε‖∇ϕ‖[L2(Ω)]n + C√ε‖∇ϕ‖[L2(Ω̂ε,3)]n .Finally, we show that‖ε∇ϕ̃‖[(H1(Ω))′]n  Cε‖∇ϕ‖[L2(Ω)]n +C√ε‖∇ϕ‖[L2(Ω̂ε,3)]n , from which the second estimaof Theorem 2.2.3. Interior error estimateIn this sectionΩ is a bounded domain inRn with a sufficiently smooth boundary (C1,1).We consider the following homogenization problem:ϕε ∈ H 10 (Ω),∫ΩA( .ε)∇ϕε · ∇ψ =∫Ωf ψ, ∀ψ ∈ H 10 (Ω), (3)wheref belongs toL2(Ω) and whereA is a square matrix of elements belonging toL∞per(Y ), satisfying thecondition of uniform ellipticityc|ξ |2  A(y)ξ · ξ  C|ξ |2 a.e.y ∈ Y , with c andC strictly positive constants.We have shown (see Theorem 4.1 in [5]) that254 G. Griso / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 251–254r,.‖ϕε − Φ‖L2(Ω) +∥∥∥∥∇ϕε − ∇Φ − n∑i=1Qε(∂Φ∂xi)∇χi( .ε)∥∥∥∥[L2(Ω)]n  C√ε‖f ‖L2(Ω) (4)where(Φ, ϕ̂) ∈ H 10 (Ω) × L2(Ω;H 1per(Y )/R) is the solution of the limit problem of unfolding homogenization∫Ω∫YA{∇xΦ + ∇y ϕ̂} · {∇xΨ + ∇yψ̂} = ∫Ωf Ψ, ∀(Ψ, ψ̂) ∈ H 10 (Ω) × L2(Ω;H 1per(Y )/R). (5)We recall that the correctorsχi , i ∈ {1, . . . , n}, are the solutions of the variational problemsχi ∈ H 1per(Y ),∫YA(y)∇y(χi(y) + yi)∇yψ(y)dy = 0, ∀ψ ∈ H 1per(Y ).They allow us to expresŝϕ in terms of∇Φ: ϕ̂ = ∑ni=1 ∂Φ∂xi χi and to give the homogenized problem satisfied byΦ∫ΩA∇Φ∇Ψ =∫Ωf Ψ ∀Ψ ∈ H 10 (Ω) (6)whereAij = 1|Y |∑nk,l=1∫Yakl∂(yi+χi )∂yk∂(yj+χj )∂yl(see [3]).Theorem 3.1. LetΩ ′ be a subdomain such thatΩ ′ ⊂⊂ Ω . The solution of problem(3) satisfied the following erroestimate:∥∥∥∥ϕε − Φ − ε n∑i=1Qε(∂Φ∂xi)χi( .ε)∥∥∥∥H1(Ω ′) Cε‖f ‖L2(Ω). (7)The constant depends onA, onΩ ′ and onΩ .Idea of the proof. The solutionΦ of the homogenized problem(6) belongs toH 10 (Ω) ∩ H 2(Ω) and wehave‖∇Φ‖[L2(Ω̂ε,3)]n  C√ε‖f ‖L2(Ω). Using (4) we obtain‖∇ϕε‖[L2(Ω̂ε,3)]n  C√ε‖f ‖L2(Ω). Let Ψ be inH 10 (Ω) ∩ H 2(Ω). In (3) we takeΨ + ε∑ni=1 ρεQε( ∂Ψ∂xi )χi(.ε) as test function whereρε(x) = inf{dist(x,∂Ω)ε ,1}and we transform the left-hand side by unfolding.We then apply Theorem 2.2 to functionϕε. Due to the estimate of‖∇ϕε‖[L2(Ω̂ε,3)]n we obtain|∫ΩA∇(ϕε −Φ)∇Ψ |  Cε‖f ‖L2(Ω)‖Ψ ‖H2(Ω) whereA is the matrix of problem(6). This gives‖ϕε −Φ‖L2(Ω)  Cε‖f ‖L2(Ω).We now write the variational problems satisfied byρϕε and (ρΦ,ρϕ̂) where ρ(x) = dist(x, ∂Ω). Proceed-ing as in Proposition 4.4 in [5] and thanks to theL2 estimate ofϕε − Φ we obtain ‖ρ(∇ϕε − ∇Φ −∑ni=1Qε( ∂Φ∂xi )∇yχi( .ε ))‖[L2(Ω)]n  Cε‖f ‖L2(Ω), hence(7). References[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.[2] D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding andhomogenization, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104.[3] D. Cioranescu, P. Donato, An Introduction to Homogenization, in: Oxford Lecture Series in Math. Appl., vol. 17, Oxford University Press1999.[4] G. Griso, Estimation d’erreur et éclatement en homogénéisation périodique, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 333–336.[5] G. Griso, Error estimate and unfolding for period c homogenization, Asymptotic Anal., in press.[6] O.A. Oleinik, A.S. Shamaev, G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992

Interior error estimate for periodic homogenization

Crossref

INTERIOR ERROR ESTIMATE FOR PERIODIC HOMOGENIZATION

International audienceIn a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\varepsilon^{1/2}$. Now, for an open set with su ciently smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions we show that in any open set strongly included in the error is of order $\varepsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or polyhedral (n=3) boundary we also give the global and interrior error estimates

Interior error estimate for periodic homogenization

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Crossref

Hal-Diderot

Comptes Rendus Mathématique