Grégory Vial

Monique Dauge

Sébastien Tordeux

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–642http://france.elsevier.com/direct/CRASS1/Partial Differential EquationsMatching and multiscale expansions for a model singularperturbation problemSébastien Tordeux a, Grégory Vial b, Monique Dauge ca MIP, INSA Toulouse, 31077 Toulouse, Franceb IRMAR, ENS Cachan Bretagne, 35170 Bruz, Francec IRMAR, Université de Rennes 1, 35042 Rennes, FranceReceived 21 August 2006; accepted after revision 12 October 2006Available online 7 November 2006Presented by Alain BensoussanAbstractWe consider the Laplace–Dirichlet equation in a polygonal domain which is perturbed at the scale ε near one of its vertices.We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of ε. On the base of thismodel problem, we compare two different approaches: the method of matched asymptotic expansions and the method of multiscaleexpansion. We enlighten the specificities of both techniques, and show how to switch from one expansion to the other. To cite thisarticle: S. Tordeux et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméDéveloppements raccordé et multi-échelle pour un problème de perturbation singulière modèle. On considère le problèmede Laplace–Dirichlet dans un domaine polygonal qui présente une perturbation de taille ε en l’un de ses sommets. Cette perturbationest supposée auto-similaire, i.e. provient d’un motif fixe dilaté à l’échelle ε. Sur ce problème modèle, nous mettons en œuvre deuxméthodes : développements asymptotiques raccordés et développement multi-échelle. Nous mettons en évidence les particularitésde chaque approche et montrons comment passer d’un développement à l’autre. Pour citer cet article : S. Tordeux et al., C. R.Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeOn considère un polygone ω de R2, qu’on perturbe localement au voisinage d’un de ses sommets, O placé àl’origine du repère. Le domaine ω coïncide au voisinage de O avec un secteur infini K dont l’angle associé est noté α.La perturbation est auto-similaire : elle provient d’un motif Ω qui est un domaine infini coïncidant avec le mêmesecteur K à l’infini ; le domaine perturbé est défini comme :ωε ={x ∈ ω; |x| > εR∗} ∪ {x ∈ εΩ; |x| < r∗}, (1)E-mail addresses: sebastien.tordeux@insa-toulouse.fr (S. Tordeux), gvial@bretagne.ens-cachan.fr (G. Vial), monique.dauge@univ-rennes1.fr(M. Dauge).1631-073X/$ – see front matter © 2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2006.10.010638 S. Tordeux et al. / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–642pour des r∗ et R∗ convenables, voir Fig. 1. Nous nous intéressons au problème modèle suivant :Trouver uε ∈ H10(ωε) tel que − uε = f dans ωε, (2)où le second membre f ∈ L2(ωε) a son support disjoint de la zone de perturbation.Les singularités du problème de Dirichlet dans le secteur K jouent un rôle essentiel dans la structure de uε . Rappe-lons qu’avec la notation λ = π/α et les coordonnées polaires (r, θ), ces singularités sont les spλ = rpλ sin(pλθ) pourtout entier relatif p [3,6].La technique de développement asymptotique multi-échelle [7,1] consiste à construire une approximation globalede uε dans ωε , composée de deux types de termes : les premiers interviennent en la variable standard x et les secondsen la variable dilatée x/ε. Les termes sont superposés à l’aide de fonctions de troncature (χ est nulle proche dupoint O , ψ est nulle à l’infini – voir (11) pour plus de détails) :uε(x) = χ(xε) n∑=0ελvλ(x) + ψ(x)n∑=0ελV λ(xε)+O(ε(n+1)λ). (3)L’approche des développements asymptotiques raccordés [4,5] consiste à construire un développement intérieur, va-lide localement autour de la perturbation, et un autre, extérieur, valide seulement loin de la perturbation. Ils doiventêtre raccordés dans la région intermédiaire. On utilise une fonction de troncature ϕ à l’échelle intermédiaire r/√εpour obtenir une approximation globale :uε(x) = ϕ(r√ε) n∑=0ελuλ(r, θ) +(1 − ϕ(r√ε)) n∑=0ελUλ(rε, θ)+O(ε(n+1)λ/2). (4)Chaque méthode a ses avantages et ses inconvénients, pour les principaux : la méthode multi-échelle présentel’avantage d’une approximation globale avec estimation optimale du reste, ainsi que le fait que tous les termes dudéveloppement sont solutions de problèmes variationnels, alors que la méthode de raccord permet de définir destermes intrinsèques, indépendamment de toute fonction de troncature. Par ailleurs, on peut passer de l’un à l’autre :il suffit de retrancher à unλ ou à Unλ une combinaison linéaire de fonctions singulières duales s−pλ, p > 0, pourretrouver les termes variationnels vnλ ou V nλ.Les détails des preuves des résultats annoncés sont exposés dans [8].1. IntroductionWe consider families of self-similar perturbed domains defined thanks to two domains ω and Ω of R2, satisfyingthe following conditions for suitable positive numbers r∗ and R∗:– ω is a bounded polygon with one vertex at the origin O: inside the ball centered in O of radius r∗, ω coincideswith an infinite plane sector K of opening α;– Ω is an unbounded domain, coinciding with K outside the ball centered in O of radius R∗.The domains ωε are defined as perturbations of ω using the pattern Ω : For ε small enoughωε ={x ∈ ω; |x| > εR∗} ∪ {x ∈ εΩ; |x| < r∗}. (5)Fig. 1 shows an example of such a situation: the corner is ‘rounded’ at vertex O to the scale ε. Of course, ourgeometrical setting covers a wider range of cases, and the domain Ω may have corners itself, or even cracks. Besides,we do not need any assumption of inclusion of ωε into ω (or conversely). The domain ωε tends to ω as ε goes to 0,and Ω appears to be the limit of the family of domains ε−1ωε .We denote by uε the solution of the Laplace–Dirichlet equation in ωε:uε ∈ H10(ωε); −uε = f in ωε, (6)where f is an L2-function whose support does not reach the origin O . The solution uε clearly converges towardu0 ∈ H10(ω), satisfying −u0 = f in ω. We aim at describing precisely the asymptotic behavior of uε by means of(i) a multi-scale expansion, and (ii) matched asymptotic expansions. We give in §2 the first terms of these expansions,S. Tordeux et al. / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–642 639Fig. 1. Example of domains ω, Ω and ωε .the complete expansions being described in §3 and §4, and compared with each other in §5. Details and proofs for thestated results can be found in [8].The multi-scale approach of this problem has already been studied within a very general framework in [7]. Ourpurpose is to give for this model problem a simple explicit presentation for the two distinct, and even competing,techniques of multi-scale and matched expansions. Besides, this allows to prove formulas for transforming eachexpansion into the other one.Notation. We need some notation used throughout the Note.• Singular exponents and functions associated with the Laplace–Dirichlet problem in the sector K are denoted bypλ and spλ, respectively, with:λ = πα, p ∈ Z, p = 0 and spλ(r, θ) = rpλ sin(λθ), with (r, θ) polar coordinates in K.• Remainders will be measured in the energy norm: we will write OH1(εκ), which means that the norm in H1(ωε)is uniformly bounded by Cεκ as ε → 0.2. The first terms of the expansionsWe present here the principles of the two methods, and provide the first terms of each expansion.2.1. Multi-Scale Expansion (MSE)The MSE method consists in looking for an expansion of uε in powers of ε with ‘coefficients’ combining the twoscales x and xε. Let χ be a smooth cut-off function vanishing at point O and ψ a smooth cut-off function localizednear O . As a result of the MSE method we find for the first terms, see Theorem 1uε = χ(xε)v0(x) + ψ(x)ελV λ(xε)+OH1(ε2λ), (7)and, nextuε = χ(xε)(v0(x) + ε2λv2λ(x)) + ψ(x)(ελV λ(xε)+ ε2λV 2λ(xε))+OH1(ε3λ). (8)Here, the first term v0 coincides with the limit u0. The profiles V λ and V 2λ are defined in the infinite patterndomain Ω . Thus information concerning the perturbing pattern is contained in the profiles (whose contribution islocalized near O), and v0, v2λ carry information corresponding to the bounded domain ω (whose influence does notreach the corner).In our MSE analysis, all the terms v(x) and all the profiles V (X) are solution of variational problems in ω and Ω ,respectively.640 S. Tordeux et al. / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–6422.2. Matched Asymptotic Expansions (MAEs)The MAEs method consists in constructing two full different expansions (outer and inner expansions, cf [4]) of uεin powers of ε. The coefficients of the outer expansion are functions of the slow variable x, and those of the inner oneare functions of the rapid variable x/ε. The outer expansion is valid far from O and the inner one in a layer close to O .But neither of these two expansions is valid everywhere. They have to be matched and glued inside an intermediateregion.In order to have a representation of uε everywhere and to optimize remainders, we use a cut-off function at theintermediate scale r/√ε. Let ϕ be a smooth cut-off function with ϕ(ρ) = 0 for ρ  1 and ϕ(ρ) = 1 for ρ  2. As aresult of the MAEs method we find for the first terms, see Theorem 2,uε = ϕ(r√ε)u0(x) +(1 − ϕ(r√ε))ελUλ(xε)+OH1(ελ), (9)and, nextuε = ϕ(r√ε)(u0(x) + ε2λu2λ(x)) + (1 − ϕ( r√ε))(ελUλ(xε)+ ε2λU2λ(xε))+OH1(ε3λ/2). (10)Here, again, the first term u0 coincides with the limit u0. The term u2λ is defined on ω whereas the profiles Uλ andU2λ are defined in the infinite domain Ω . The terms uλ(x), Uλ(X) and U2λ(X) are solution of ‘super-variationalproblems’, i.e. problems set in spaces larger than the variational spaces, and where standard formulations have non-unique solutions. Moreover, the terms upλ and Upλ are, in general, distinct from the vpλ and V pλ and differ by acombination of dual singular functions, see Theorem 3.3. Multiscale expansionIn the MSE approach, the global expansion is valid everywhere and consists of terms involving the two scales xand x/ε, superposed via cut-off functions: χ and ψ are smooth and radial, satisfying:{χ(X) = 1 for |X| > 2R∗ and χ(X) = 0 for |X| < R∗,ψ(x) = 1 for |x| < r∗2and ψ(x) = 0 for |x| > r∗. (11)Theorem 1. The solution uε of problem (6) admits the following multiscale expansion into powers of εuε(x) = χ(xε) n∑=0ελvλ(x) + ψ(x)n∑=0ελV λ(xε)+OH1(ε(n+1)λ), (12)where the terms vλ and V λ do not depend on ε, and are defined in ω and Ω by Eqs. (13) and (14).3.1. Short description of the termsGenerically, vλ and V λ solve the variational problemsvλ ∈ H10(ω) such that ∀v′ ∈ H10(ω),∫ω∇vλ · ∇v′ =∫ωf λv′, (13)V λ ∈ W10(Ω) such that ∀V ′ ∈ W10(Ω),∫Ω∇V λ · ∇V ′ =∫ΩFλV ′, (14)where W10(Ω) is the space {V ∈ W1(Ω); V = 0 on ∂Ω}, with the weighted space W1(Ω) defined as:W1(Ω) = {V ∈D′(Ω); (1 + R)−1V ∈ L2(Ω), ∇V ∈ L2(Ω)} with R = |X|. (15)The right-hand sides f λ and Fλ are defined recursively.S. Tordeux et al. / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–642 641• For  = 0, we have f 0 = f , defined in problem (6) (v0 is equal to the limit solution u0), and F 0 = 0;• For  = 1, the function f λ is 0 and Fλ = b0λ([X,χ]sλ) = b0λ(X(χsλ) − χX(sλ)), with the scalar coefficientb0pλ arising from the decomposition into singular functions of v0, cf. [6,3]:v0(x) =q∑p=1b0pλ spλ(r, θ) + Or→0(rqλ), ∀x ∈ ω, ∀q ∈ N; (16)• The next terms are generically nonzero: f λ comes from the expansion at infinity of the previous profiles V pλ(p < ), namely for the first oneV λ(X) =q∑p=1Bλpλ s−pλ(R, θ) + OR→∞(R−qλ), ∀X ∈ Ω, ∀q ∈ N, (17)and Fλ from the decomposition into singular functions of the previous terms vpλ (p < ).4. Matched asymptotic expansionsLet ϕ be a smooth cut-off with ϕ(ρ) = 0 for ρ  1 and ϕ(ρ) = 1 for ρ  2. The outer and inner expansions of uεare glued together with the cut-off function x → ϕ(r/√ε):Theorem 2. The solution uε of problem (6) admits the following expansion into powers of εuε(x) = ϕ(r√ε) n∑=0ελuλ(r, θ) +(1 − ϕ(r√ε)) n∑=0ελUλ(rε, θ)+OH1(ε(n+1)λ/2), (18)where the terms uλ and Uλ do not depend on ε, and are defined in ω and Ω by Eqs. (19) and (20).4.1. Short description of the termsGenerically, uλ and Uλ solve ‘super-variational’ problems:Find uλ ∈ Vloc,0(ω) such that uλ = −δ0f in ω and uλ −−1∑p=1aλpλ s−pλ ∈ H1(ω), (19)Find Uλ ∈ Vloc,∞(Ω) such that Uλ = 0 in Ω and Uλ −∑p=1Aλpλ spλ ∈ W1(Ω), (20)where the spaces Vloc,0(ω) and Vloc,∞(Ω) are defined asVloc,0(ω) ={u ∈D′(ω); Φu ∈ H10(ω), ∀Φ ∈ C∞(ω̄) with O /∈ supp(Φ)}, (21)Vloc,∞(Ω) ={U ∈D′(Ω); ΦU ∈ H10(Ω), ∀Φ ∈ C∞(Ω) with compact support}. (22)In (19) δ0 is the Kronecker symbol. The coefficients aλpλ and Aλpλ are constructed by induction, and play also therole of right-hand sides: In these super-variational problems, uniqueness is enforced by prescribing an asymptoticsin terms of dual singular functions. The coefficients aλpλ and Aλpλ come from the decomposition into singular anddual-singular functions of the terms Upλ and upλ for p < , respectively.Remark 1. The scale r/√ε used in expansion (18) is, in a certain sense, arbitrary. If, instead, we choose the scale x/εγwith γ ∈ (0,1), we find for the remainder OH1(ε(n+1)λmin{γ,1−γ }). Thus the scale r/√ε optimizes the remainder.642 S. Tordeux et al. / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 637–6425. Comparison between the two expansionsTheorem 3. The expansions (12) and (18) can be compared as follows: the terms unλ and vnλ coincide away from thecorner point i.e. for r  r∗; the profiles Unλ and V nλ coincide in the corner region i.e. for R  R∗/2.More precisely, we have the identities ( for the definition of the coefficients anλpλ and Anλpλ, see Section 4)⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩vnλ(x) = unλ(x) − ψ(x)n−1∑p=1anλpλ s−pλ(x), x ∈ ω,V nλ(X) = Unλ(X) − χ(X)n∑p=1Anλpλ spλ(X), X ∈ Ω.(23)6. ExtensionsSeveral extensions of the previous results are possible, requiring more or less effort. First, we can consider asmooth right-hand side f whose support does reach the perturbation region. In this case, the so-called logarithmic-polynomial singularities have to be considered to take the Taylor expansion of f near the corner point O into account.Other boundary conditions may be treated as well. The Neumann case, for instance, is slightly more involved due tothe logarithmic singularity and the absence of Poincaré inequality. Lastly, we mention the three-dimensional case: ifthe domain ω has only a conical point at O , the analysis is very similar to the two-dimensional case. However, inpresence of edges, these techniques might still be used, but with non straightforward adaptations (see [2] for edge andcorner-edge asymptotics without small parameters).References[1] G. Caloz, M. Costabel, M. Dauge, G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer,Asymptotic Analysis (2006), in press.[2] M. Dauge, Elliptic Boundary Value Problems in Corner Domains – Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics,vol. 1341, Springer-Verlag, Berlin, 1988.[3] P. Grisvard, Boundary Value Problems in Non-Smooth Domains, Pitman, London, 1985.[4] A. Il’lin, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, 1992.[5] P. Joly, S. Tordeux, Matching of asymptotic expansions for wave propagation in media with thin slots I: The asymptotic expansion, MultiscaleModeling and Simulation 5 (1) (2006) 304–336.[6] V.A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16(1967) 227–313.[7] V. Maz’ya, S.A. Nazarov, B.A. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains,Birkhäuser, Berlin, 2000.[8] S. Tordeux, G. Vial, M. Dauge, Selfsimilar perturbation near a corner: matching and multiscale expansions, 2006, in preparation.

Matching and multiscale expansions for a model singular perturbation problem

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Matching and multiscale expansions for a model singular perturbation problem

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