Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

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1 wo-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor Ionel Sorin Ciuperca, Ronan Perrussel, Clair Poignard o cite this version: Ionel Sorin Ciuperca, Ronan Perrussel, Clair Poignard. wo-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor. Journal de Mathématiques Pures et Appliquées, Elsevier, 211, J Math Pures Appl 95(3):19 (211) 95 (3), pp <1.116/j.matpur >. <inria-41835> HAL Id: inria Submitted on 6 Jul 29 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. he documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 INSIU NAIONAL DE RECHERCHE EN INFORMAIQUE E EN AUOMAIQUE wo-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor Ionel Ciuperca Ronan Perrussel Clair Poignard N 6975 Juin 29 hème NUM apport de recherche ISSN ISRN INRIA/RR FR+ENG

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4 wo-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor Ionel Ciuperca, Ronan Perrussel, Clair Poignard hème NUM Systèmes numériques Équipes-Projets MC2 Rapport de recherche n 6975 Juin pages Abstract: We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. he roughness of the layer is described by a quasi periodic function, being a small parameter, while the mean thickness of the layer is of magnitude β, where β (, 1). Using the twoscale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 23; 37: ). his paper extends the previous works Poignard (Math. Meth. App. Sci. 29; 32: ) and Ciuperca et al. (Research report INRIA RR-6812), in which β 1. Key-words: Asymptotic analysis, Finite Element Method, Laplace equations Université de Lyon, Université Lyon 1, CNRS, UMR 528, Institut Camille Jordan, Bat. Braconnier, 43 boulevard du 11 novembre 1918, F Villeurbanne Cedex, France Laboratoire Ampère UMR CNRS 55, Universit de Lyon, École Centrale de Lyon, F Écully, France INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 & Université de Bordeaux1, 351 cours de la Libération, 3345 alence Cedex, France Centre de recherche INRIA Bordeaux Sud Ouest Domaine Universitaire - 351, cours de la Libération 3345 alence Cedex éléphone :

5 Analyse double échelle pour des couches minces très rugueuses. Une caractérisation explicite du tenseur de polarisation Résumé : Mots-clés : Analyse Asymptotique, Méthode des Eléments Finis, Equations de Laplace

6 Approximate transmission conditions for very rough thin layers 3 Contents 1 Introduction Description of the geometry Statement of the problem Main results Variational formulations Approximate transmission conditions Some preliminary results Preliminary estimates Change of variables First convergence results Computation of the limit of wo-scale convergence of β s z and t z Proofs of heorem 2.3 and heorem a Proof of heorem b Proof of heorem Conclusion 18 RR n 6975

7 4 Ciuperca& Perrussel & Poignard m 1 Γ Γ 1 Introduction Figure 1: Geometry of the problem. Consider a material composed of a two-dimensional object surrounded by a very rough thin layer. We study the asymptotic behaviour of the steady-state voltage potential when the thickness of the layer tends to zero. We present approximate transmission conditions to take into account the effects due to the layer without fully modeling it. his paper ends a series of 3 papers dealing with the steady-state voltage potential in domains with thin layer with a non constant thickness. Unlike [16, 17] in which the layer is weakly oscillating, and unlike [11], which deals with the periodic roughness case, we consider here the case of a very rough thin layer. his means that the period of the oscillations is much smaller than the mean thickness of the layer. More precisely, we consider a period equal to, while the mean thickness of the layer is of magnitude β, where β is a positive constant strictly smaller than 1. As for [11], the motivation comes from a collaborative research on the modeling of silty soil, however we are confident that our result is useful for more different applications, particularly in the electromagnetic research area. 1.1 Description of the geometry For sake of simplicity, we deal with the two-dimensional case, however the threedimensional case can be studied in the same way up to few appropriate modifications. Let be a bounded smooth domain of R 2 with connected boundary. For >, we split into three subdomains: 1, m and. 1 is a smooth domain strictly embedded in. We denote by Γ its connected boundary. he domain m is the thin oscillating layer surrounding 1 (see Fig. 1). We denote INRIA

8 Approximate transmission conditions for very rough thin layers 5 by Γ the oscillating boundary of m : Γ = m \ Γ. he domain is defined by We also write = \ ( 1 m ). = \ 1. We suppose that the curve Γ is a smooth closed curve of R 2 of length 1, which is parametrized by its curvilinear coordinate: { } Γ = γ(t), t, where is the torus R/Z. Denote by ν the normal to Γ outwardly directed to 1. he rough boundary Γ is defined by Γ = {γ (t), t }, where γ (t) = γ(t) + β f ( t, t ) ν(t), where < β < 1 and f is a smooth, (1, 1) periodic and positive function such that 1 2 f 3 2. Observe that the membrane has a fast oscillation compared with the size β of the perturbation. 1.2 Statement of the problem Define the piecewise regular function σ by σ 1, if x 1, x, σ (x) = σ m, if x m, σ, if x, where σ 1, σ m and σ are given positive 1 constants and let σ : R be defined by 2 { σ 1, if x 1, σ(x) = σ, if x. Let g belong to H s (), for s 1. We consider the unique solution u to. (σ u ) =, in, (1a) u = g. Let u be the unique solution to the limit problem (1b). (σ u) =, in, (2a) u = g. (2b) 1 he same following results are obtained if σ 1, σ m and σ are given complex and regular functions with imaginary parts (and respectively real parts) with the same sign. 2 σ represents the piecewise-constant conductivity of the whole domain. RR n 6975

9 6 Ciuperca& Perrussel & Poignard Since the domains, 1 and are smooth, the above function u belongs to H s ( 1 ) and H s ( ). In the following we suppose that s > 3 hence by Sobolev embeddings there exists s > such that u C 1,s ( 1 ) and u C 1,s ( ). We aim to give the first two terms of the asymptotic expansion of u for tending to zero. Several papers are devoted to the modeling of thin layers: see for instance [8, 7, 16] for smooth thin layers and [1, 2, 4, 14, 11] for rough layers. However, as far as we know, the case of very rough thin layer has not been treated yet. In [1] Vogelius and Capdeboscq derive a general representation formula of the steady-state potential in the very general framework of inhomogeneities of low volume fraction, including the case of very rough thin layers. However their result involves the polarization tensor, which is not precisely given. his paper can be seen as an explicit characterization of the polarization tensor for very rough thin layers. Our main result (see heorem 2.3) is weaker than the results of [16, 11], since we do not prove error estimates. Actually, using variational techniques we prove that the sequence (u u)/ β weakly converges in L p (), for all p (1, 2) to a function z. his function z is uniquely determined by the elliptic problem (11), and the convergence does hold in L s, for s 1 far from the layer (see heorem 2.7). In the present paper it seems difficult to obtain the H 1 strong convergence in as in [11]. he main reason comes from the fact that according to Bonder et al., the best Sobolev trace constant blows up for tending to zero in the case of a very rough layer. herefore, the analysis performed previously can not be applied. o obtain our present result, we use a variational technique based on the two-scale analysis. We emphasize that this technique can be applied to obtain the limit problems presented in [16, 11], even if the error estimates are more complex to be achieved in such a way. We conclude by observing that the two-scale convergence enables us to draw the target to be reached: another asymptotic analysis as to be performed to obtain error estimates, however the result is sketched. he outline of the paper is the following. In the next section we present precisely our main results using a variational formulation. Section 3 is devoted to preliminary results. In particular, we show the first two limits easy to be reached. In Section 4, we end the proof of the main theorems by computing the limit of E defined by (19). We then conclude the paper with numerical simulations, which illustrate the theoretical results. We shall first present our main results. 2 Main results 2.1 Variational formulations Denote by z the element of H 1 () defined by z = u u β. INRIA

10 Approximate transmission conditions for very rough thin layers 7 We shall obtain the limit of z with the help of variational techniques. Since g belongs to H s (), for s > 3, we define by g + H 1 () the affine space } g + H() 1 = {v H 1 () : v = g. he variational formulation of Problem (1) is Find u g + H() 1 such that: σ u ϕ =, and respectively for Problem (2) Find u g + H() 1 such that: σ u ϕ =, ϕ H 1 (), ϕ H 1 (). aking the difference between the above equalities, z belongs to H 1 () and satisfies σ z ϕ = 1 β (σ σ) u ϕ, ϕ H 1 (), (3) or equivalently σ z ϕ = (σ σ) z ϕ 1 β (σ σ) u ϕ, ϕ H 1 (). (4) Notation 2.1 (Normal and tangential derivatives). Denote by θ(t) the tangent vector to Γ in any point γ(t): t, θ(t) = (γ 1 (t), γ 2 (t)). he normal vector ν outwardly directed to 1 is then given by t, ν(t) = (ν 1 (t), ν 2 (t)) = (γ 2 (t), γ 1 (t)). In the following, for any x Γ and for any function ϕ smooth enough, we denote the normal and tangential derivatives of ϕ respectively by ϕ + ν (x) = lim ϕ ϕ(y) ν, y x,y ϕ (x) = ϕ(x) θ. θ We also write ϕ + (x) = ν (x) = lim ϕ(y) ν, y x,y 1 lim y x,y ϕ(y), ϕ (x) = lim y x,y 1 ϕ(y). Notation 2.2 (Green operator). We introduce the Green operator G : H 1 () H 1 () given by G(ψ) = ϕ iff ϕ is the unique solution of the problem. (σ ϕ) = ψ in, (5a) ϕ =. (5b) It is well known that if ψ L p () with p > 2 then ϕ W 2,p ( k ), k =, 1, then by Sobolev embeddings there exists s > such that ϕ C 1,s ( 1 ) and ϕ C 1,s ( ). RR n 6975

11 8 Ciuperca& Perrussel & Poignard 2.2 Approximate transmission conditions Let f min and f max be f min = min f(t, τ) and f max = max f(t, τ). t,τ t,τ For sake of simplicity, we suppose that 1 2 f min f max 3 2. For any fixed t and s R we denote by Q(s, t) the one-dimensional set (s, t) R, Q(s, t) = {τ, s f(t, τ)}, and let q(s, t) be the Lebesgue-measure of Q(s, t): (s, t) R, q(s, t) = χ Q(s,t) (τ)dτ, (6) where χ A is the characteristic function of the set A. Observe that q satisfies q(s, t) 1, q(s, t) = 1 for s < f min and q(s, t) = for s > f max. Moreover since q is a measurable function it belongs to L. We also write f(t) = 1 f(t, τ)dτ. (7) Our approximate transmission conditions need the two following functions t, r 1 (t) = t, r 2 (t) = fmax fmax f min q 2 (s, t) ds, (8) σ m (γ(t))q(s, t) + σ (γ(t))[1 q(s, t)] q(s, t)[1 q(s, t)] ds. (9) σ (γ(t))q(s, t) + σ m (γ(t))[1 q(s, t)] o simplify notations, we still denote by r k the function of Γ equal to r k γ 1, for k = 1, 2. he aim of the paper is to prove the following theorem. heorem 2.3 (Main result). here exists z 1<p<2 L p () such that z weakly converges to z in L p () for all p (1, 2). he limit z is the unique solution to ψ p >2L p (), [ ( ) u + ϕ zψ dx = (σ σ m ) f + ] + (σ σ m )r 1 dγ Γ ν ν [ ( ) ] u ϕ + (σ σ m ) f + (σ σ m )r 2 dγ, θ θ Γ (1) where ϕ = G(ψ). Remark 2.4. he existence and the uniqueness of z 1<p<2 L p () solution of (1) comes from the fact that for any p > 2 the dual of L p () is L p () with 1/p + 1/p = 1 and that the expression of the right-hand side of (1) is a continuous linear application from L p () to R with argument ψ. INRIA

12 Approximate transmission conditions for very rough thin layers 9 Remark 2.5. From the uniqueness of z we deduce that the whole sequence z converges to z. Remark 2.6 (Strong formulation). We can write a strong formulation of (1). Supposing that z is regular enough on and on 1, and taking in (1) appropriate test functions, we infer that z satisfies the following problem. (σ k z) = in k, k =, 1, (11a) ( z + z = 1 σ ) [ ] m u + f + (σ σ m )r 1 on Γ, (11b) σ ν z + σ ν σ z 1 ν = [ ( ) ] u (σ σ m ) f + (σ σ m )r 2 on Γ, (11c) θ θ z =. (11d) Moreover, using the regularity of u in H s ( ), with s > 3, we infer easily the existence and the uniqueness of z in H s 1 ( ) and H s 1 ( 1 ). heorem 2.7 (Strong convergence far from the layer). Let D be an open set such that Γ D and D. hen the sequence z converges strongly to z in L p ( \ D), for all p 1. Remark 2.8 (he case of a thin layer with constant thickness). In the particular case where f is independent on τ, we have f = f(t) and { 1 for s f(t), q(s, t) = (12) for s f(t), and hen (11) becomes r 1 (t) = f(t) σ m (γ(t)) and r 2 (t) =.. (σ k z) = in k, k =, 1, (13a) ( ) z + z σ = 1 f u+ on Γ, (13b) σ m ν z + σ ν σ z 1 ν = ( f(σ σ m ) u ) on Γ, (13c) θ θ z =. (13d) which is the result obtained in [16, 17]. 3 Some preliminary results 3.1 Preliminary estimates Lemma 3.1. he following estimates hold. i) here exists C > such that z H 1 () C β/2. ii) For any p ]1, 2[ there exists C p > such that z L p () C p. RR n 6975

13 1 Ciuperca& Perrussel & Poignard Proof. i): ake ϕ = z in (3) and use the regularity of u. ii): For any p ]1, 2[ we introduce the function z p defined on by z p (x) = z (x) z (x) p 2 χ {z(x) }. We have z p z = z p. hen we take ϕ = G(z p ) as a test function in (4); in the left-hand side we obtain z p L p (). Let p 1 = p p 1 > 2, then ϕ L () C p z p L p 1() = z p 1 L p (), and using i) we easily see that the right-hand side of (4) can be bounded by a term like C z p 1 L p (). his gives the result. 3.2 Change of variables We shall use the change of variables: where α : R R 2 is an application given by x = α (s, t), (14) α (s, t) = γ(t) + β sν(t). Denote by κ the curvature 3 of Γ. For >, we denote by C the rough cylinder Let d be such that C = {(s, t), t, s f(t, t/)}. < d < 1 κ. (15) For all (, d 1/β ), α is a diffeomorphism between the rough cylinder C and m. he Jacobian matrix A of α equals where (s, t) ( 1, 1), A (s, t) = J (t) ( ν1 (t) ν t, J (t) = 2 (t) ν 2 (t) ν 1 (t) ( β 1 + β sκ(t) According to (15), A is invertible. Denote by B its inverse matrix ( ) β (s, t) ( 1, 1), B (s, t) = 1/(1 + β J sκ(t)) (t). For any functions v and w belonging to H 1 (R 2 ), define the functions v and w by (s, t) ( 1, 1), v(s, t) = v α (s, t), w(s, t) = w α (s, t). 3 κ is the function defined by t, ν (t) = κ(t)γ (t). ). ), INRIA

14 Approximate transmission conditions for very rough thin layers 11 Let s,t be the gradient operator ( s, t ). Using the change of variables, and since J = J 1 we obviously have on (, 2) ( x v x w) α = ( s,t v) B (B ) s,t w, = 1 2β 1 sv s w + (1 + β sκ) 2 tv t w. (16) Hence x v α x w α is close to v w v w t t + 2β s s 3.3 First convergence results on (, 2). For any fixed ψ p >2L p () we take ϕ = G(ψ) as a test function in (4). We obtain z ψ dx = (σ σ m )(E + E ), (17) where E = 1 β u ϕdx, (18) m E = z ϕ. (19) m We pass to the limit in the left-hand side of (17) thanks to Lemma 3.1. Up to an appropriate subsequence we infer z ψ dx = zψ dx. (2) lim he aim of the paper is to obtain the limits of E and E. It is easy to compute the limit of E. Actually, using the change of variables (s, t) in the expression of E we infer, for ǫ small enough 4, s (,f max) E = f(t,t/) he regularity of u and ϕ implies that ( sup u α u ϕ (s,.) ϕ α (s,.) ν ν (1 + β sκ(t)) u α (s, t) ϕ α (s, t)ds dt. (21) γ + + u θ ϕ θ γ+) L 2 () = O( β ). We then deduce from the weak convergence of f(t, t ) to f the limit of E : ( u + lim E Γ = ϕ + ν ν + u ) ϕ f dσ Γ. (22) θ θ herefore we have proved that up to a subsequence ( u (σ σ m ) lim E + = ϕ + zψ (σ σ m ) Γ ν ν + u ) ϕ f dσ Γ. (23) θ θ o end the proof of heorem 2.3, it remains to determine the limit of E. 4 i.e. such that ǫ β < (d /f max). RR n 6975

15 12 Ciuperca& Perrussel & Poignard 4 Computation of the limit of E he limit of E is more complex to be achieved. Now for simplicity we still denote by z the composition z α. Using the change of variables (s, t) we infer: E = β f(t,t/) (1 + β sκ) ( 1 2β sz s ϕ + ) 1 (1 + β sκ) 2 tz t ϕ ds dt. Unlike for E, the derivatives of z inside the brackets do not converge strongly. In the following, we show that for all M > f max these derivatives two-scale converge in the cylinder P M = ( M, M), for tending to zero such that β d /M. Denote by M the tubular neighbourhood of Γ composed by the points at the distance smaller than β M of Γ. By definition, α is a diffeomorphism from P M onto M and α (P M ) contains m. According to Lemma 4.1, in order to obtain the limit of E we just have to prove the two-scale convergence of the derivatives of z in P M. Actually we have the following general result on the two-scale convergence. Lemma 4.1. Let M > f max. Let v be a bounded sequence in L 2 (P M ) and let v L 2 (P M 2 ) be a two-scale limit of v for tending to zero such that β < d /M. Let also φ be a regular enough function, defined on P M. hen we have lim f(t,t/) ( v φ s, t, t ) ds dt = f(t,τ) 2 v φ(s, t, τ)dτ dy ds dt. Proof. Denote by b(s, t, τ) = φ(s, t, τ)χ {<s<f(t,τ)} defined on the set P M, which is independent on. he difficulty comes from the fact that the function b is not regular in τ, so we can not take it directly as a test function in the twoscale convergence. Using the change of variables s = rf(t, t ) with r [, 1], we infer f(t,t/) (s, φ t, t ) 2 dsdt = 1 (rf φ ( t, t ), t, t ) 2 ( f t, t ) drdt. By regularity, this last integral converges, when tends to to 1 f(t,τ) φ(rf(t, τ), t, τ) 2 [f(t, τ)] dτdrdt = φ(s, t, τ) 2 dτdsdt. We thus proved the following result: (s, b t, t ) 2 dtds P M P M b(s, t, τ) 2 dτdsdt for. (24) We similarly prove that for any φ 1 belonging to L 2 (P M, C()) we have 5 ( b s, t, t ) ( φ 1 s, t, t ) dtds b(s, t, τ)φ 1 (s, t, τ)dτdsdt for. P M P M (25) 5 We can interpret (25) as a result of partial two-scale convergence of b(s, t, t ) to b(s, t, τ). Moreover (24) says that this two-scale convergence is strong. INRIA

16 Approximate transmission conditions for very rough thin layers 13 By simply adapting the proof of heorem 11 of Lukassen et al. [15] (see also Allaire [3], heorem 1.8) we prove that the convergences (24) and (25) imply ( lim v b s, t, t ) ds dt = v b(s, t, τ)dτ dy ds dt, P M 2 which is the desired result. P M 4.1 wo-scale convergence of β s z and t z Prove now the two-scale convergence of the derivatives of z. Lemma 4.2. Let p (1, 2). here exist two constants C and C p such that for any M > 2, for any < β < d /M, we have i) z t + z β L2 (P M) s C β. L2 (P M) ii) z L p (P M) C p β/p. Proof. According to Lemma 3.1 and with the help of the change of variables (14) we straightforwardly obtain (ii). For (i) we use the formula (16) with v = w = z. By two-scale convergence there exist a subsequence of still denoted by and ξ M k (s, t, τ, y) L2 (P M ], 1[ 2 ), k = 1, 2, such that and z s ξm 1 in P M, β z t ξm 2 in P M, where denotes the two-scale convergence. For k = 1, 2 let ˆξ k M(s, t, τ) = 1 (s, t, τ, y)dy, which are functions defined on ξm k the domain P M. he following estimate is obvious: C >, M > 2, ˆξ k M Moreover if M 1 < M 2 then the restriction of ˆξ M1 k for k = 1, 2. C, k = 1, 2. (26) L2 (P M ],1[) ˆξ M2 k Lemma 4.3. For any M > f max the following results hold. i) ˆξM 1 is independent on τ. 1 ii) ˆξ M 2 dτ = a.e. (s, t). to the set { s M 1 } is exactly Proof. i) Consider θ 1 (s, t, τ) and θ 2 (s, t, τ) in D (P M ) arbitrary, such that θ 1 s + θ 2 =. (27) τ Using the two-scale convergence and also the fact that β < 1, we infer [ ( z s θ 1 s, t, t ) + z ( t θ 2 s, t, t )] 1 ˆξ 1 M θ 1, for. P M RR n 6975 P M

17 14 Ciuperca& Perrussel & Poignard On the other hand, by Green formula and according to (27) and to Lemma 4.2(ii): [ ( z s θ 1 s, t, t ) + z ( t θ 2 s, t, t )] ( θ 2 = z s, t, t ), for. P M t P M We then infer P M ˆξ M 1 θ 1 =, for any (θ 1, θ 2 )satisfying (27). Using now the De Rham theorem, we deduce that the vector (ˆξ 1 M, ) is a gradient in the variables (s, τ). Hence there exists a function H such that H s = ˆξ M 1 and H τ =, which proves i). ii) From Lemma 4.2 (ii), for any p ]1, 2[ and M > fixed we have which implies β z in L p (P M ) strongly for, (28) β z t in D (P M ). On the other hand, from Lemma 4.2 (i) there exists ξ L 2 (P M ) such that, up to a subsequence of, we have By identification we obtain Since by the two-scale theory β z t ξ in D (P M ). ξ =. we infer the result. ξ = 1 1 ξ2 M dτ dy, Define now the space H 1 per, (P M) by and let H 1 per, (P M) = {ϕ H 1 (P M ), ϕ s =M = }, D = [, 2] and D = {(s, t, τ) D, s f(t, τ)}. he next lemma shows that ˆξ M 1 is independent on M, for s 2. Lemma 4.4. For any M > f max, ˆξ M 1 = (σ σ m )q σ m q + σ (1 q) u +, for s 2, ν where σ, σ m and u+ ν are evaluated in x = γ(t) and q is defined by (6). INRIA

18 Approximate transmission conditions for very rough thin layers 15 Proof. We take as test function in (3) an element ϕ H() 1 with support in α (P M ). Using the local coordinates (s, t) and (16) we infer 1 M ( ) 1 β (1 + β sκ)σ (α ) 2β 1 sz s ϕ + (1 + β sκ) 2 tz t ϕ ds dt = M (σ σ m ) f(t,t/) 1 f(t,t/) (1 + β sκ)( s,t ϕ) B x u(α )ds dt. ake in the above equality a test function ϕ(s, t) which is an element of Hper, 1 (P M) and multiply by β. Observe that J x u(γ) = ( u u ν (γ), θ (γ)) hence [ 1 z ϕ 1 f(t,t/) lim σ 1 M s s + z ϕ σ m s s 1 ] M z ϕ 1 f(t,t/) ϕ u + + σ = (σ σ m ) lim s s s ν γ(t) +. (29) According to Lemma 4.1 with v = z and Φ in appropriate manner (for s ϕ example for the second integral we take Φ(s, t, τ) = σ m s (s, t)), we infer 1 1 ϕ σ 1ˆξM 1 M s + ϕ 1 1 M σ mˆξm 1 D s + ϕ σ ˆξM 1 f(t,τ) s = (3) ϕ u + (σ σ m ) s ν. D Let ϕ be arbitrary such that ϕ = for s f max. We deduce that ˆξ 1 M is independent on s for s f max. On the other hand, according to (26), the L 2 norm of ˆξ 1 M is uniformly bounded in M hence ˆξ M 1 =, for s f max. (31) Now choose ϕ Hper, 1 (P M) arbitrary such that ϕ = for s or s 2. Integrating (3) first in τ and using the independence of ˆξ 1 M on τ, we obtain 2 [σ m q + σ (1 q)]ˆξ 1 M ϕ 2 ds dt = (σ σ m ) u+ s ν q ϕ ds dt, s which gives s [ (σ m q + σ (1 q)) ˆξ M 1 ] = s aking into account (31) we obtain the result. ] [(σ σ m ) u+ ν q, for s 2. he next lemma gives an useful information about ˆξ M 2. Lemma 4.5. For any M > f max and any function d C() we have f(t,τ) d(t)ˆξ 2 M ds dτ dt = (σ u σ m ) θ d(t)r 2(t)dt, where r 2 is defined by (9). RR n 6975

19 16 Ciuperca& Perrussel & Poignard Proof. In (29) we take a test function ϕ in the form ϕ(s, t) = Φ(s, t, t ) where Φ is an enough regular function defined on ] M, M[ 2. Multiplying (29) by we obtain [ 1 lim M 1 M f(t,t/) σ 1 β z t σ β z t D Φ τ Φ τ ( s, t, t ) + 1 f(t,t/) ( s, t, t ) ] = (σ σ m ) lim σ m β z Φ t τ 1 f(t,t/) ( s, t, t ) + Passing to the limit and using again Lemma 4.1 we obtain 1 1 Φ σ 1ˆξM 2 M τ + Φ 1 1 M σ mˆξm 2 D τ + Φ σ ˆξM 2 f(t,τ) τ = (σ σ m ) u Φ θ τ. ( u θ (γ) Φ s, t, t ). τ (32) By density argument, this equation is also valid for Φ not regular in (s, t) but with the H 1 -regularity in τ. aking first Φ arbitrary such that Φ = for s, we deduce that ˆξ 2 M is independent on τ. With the help of Lemma 4.3(ii) we obtain ˆξ M 2 =, for s. (33) We similarly obtain ˆξ M 2 =, for s f max. (34) Let Φ be a test function such that σ m Φ τ = d(t) + c(s, t) on D, σ Φ τ = c(s, t), on D \ D, where c(s, t) must be chosen such that 1 Φ τ dτ = in order to have the periodicity in τ. Obviously, the function Φ given on D by Φ(s, t, τ) = τ ϕ 1(s, t, τ )dτ where with ϕ 1 = c(s, t) on D, σ m c(s, t) on D \ D, σ d(t) + σ m dσ q c(s, t) = σ q + σ m (1 q), (35) satisfies the required conditions. We then extend Φ on s < or s > 2 such that Φ = on s = ±M. aking this Φ as a test function in (32) and according to (33) (34) we infer: D d(t)ˆξ 2 D M + c(s, t)ˆξ 2 M = D (σ σ m ) u d + c. (36) θ σ m From Lemma 4.3 (ii) the second integral of this equality is equal to, which gives the result, according to (35). INRIA

20 Approximate transmission conditions for very rough thin layers Proofs of heorem 2.3 and heorem 2.7 We now end the proof of our main results. 4.2.a Proof of heorem 2.3 o prove heorem 2.3 it remains to compute the limit of E. Using local coordinates (s, t), E equals E = f(t,t/) Using the regularity of σ, σ m and ϕ we infer lim E = lim ( x ϕ) (α )(B ) s,t z det(a )ds dt. f(t,t/) ( x ϕ + ) (γ)j z s β z t Using now Lemma 4.1 we obtain lim E = ϕ D θ (γ)ˆξ 2 M + ϕ + D ν (γ)ˆξ 1 M. From Lemma 4.5 with d(t) = ϕ θ (γ(t)), we deduce D ϕ θ (γ)ˆξ M 2 = (σ σ m ) u θ (γ) ϕ θ (γ)r 2(t)dt. dsdt. he expression of ˆξ 1 M of Lemma 4.4 leads to ϕ + D ν (γ)ˆξ 1 M u + = (σ σ m ) ν (γ) ϕ+ ν (γ)r 1(t)dt and this last three equalities give ( u + lim E = (σ ϕ + σ m ) ν ν r 1(t) + u ) ϕ θ θ r 2(t) dγ. (37) Inserting (37) into (23) leads to equality (1) of heorem b Proof of heorem 2.7 Γ Let us show that far away from the thin layer, the sequence z is bounded in H 1. hen using a compacity argument we infer that z is the strong limit of z in L s, for all s 1, which is exactly heorem 2.7. Lemma 4.6. Let D be an open set such that Γ D and D. hen there exist two positive constants and c depending on D such that, for any ], [ we have z H1(\D) c. RR n 6975

21 18 Ciuperca& Perrussel & Poignard Proof. We proceed as in [9]. We introduce the linear operator R : H 1 ( \ D) H 1 (D) given by R(ψ) = ϕ iff ϕ is the unique solution of the problem { (σ ϕ) = in D (38) ϕ = ψ on D. It is clear, by interior regularity, that for any open set D 1 with D 1 D there exists a positive constant c 1 depending on D 1 such that R(ψ) W 1, (D 1) c 1 ψ H1 (\D), ψ H 1 ( \ D). (39) We now introduce the function ϕ defined in by { z in \ D ϕ = R(z ) in D. (4) It is clear that ϕ H() 1 so we can take it as a test function in the variational formulation (4). We obtain σ z ϕ = (σ σ) z R(z ) 1 β (σ σ) u R(z ). (41) m On the other hand, taking R(z ) z H 1 (D) as a test function in (38) with ψ = z, we obtain σ R(z ) 2 dx = σ z R(z ) so, the left-hand side of (41) becomes σ z 2 dx + D D D D m σ R(z ) 2 dx Now using i) of Lemma 3.1 and the inequality (39) we easily control the terms of the right of (41) and with the help of the Poincaré inequality on \ D we obtain the desired result. 5 Conclusion In this paper, we have derived appropriate transmission conditions to tackle the numerical difficulties inherent in the geometry of a very rough thin layer. hese transmission conditions lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq [1]. More precisely, suppose that σ = σ 1 and denote by G(x, y) the Dirichlet solution for the Laplace operator defined in [5] pp33 by ) x (σ (x) x G(x, y) = δ y, in G(x, y) =, x. According to heorem 2.7, the following equality holds almost everywhere in (u u)(y) = β x G(x, y)z(x)dx + o( β ), y. INRIA

22 Approximate transmission conditions for very rough thin layers 19 According to (11), simple calculations lead for almost every y to ( ) ( ) (u u)(y) = β n u n G (σ m σ )M(s) (s, y)d Γ u Γ G Γ (s) + o( β ), Γ where M is the polarization tensor defined by ( ) f + (σ σ s Γ, M(s) = m )r 1. f + (σ σ m )r 2 ( ) σ /σ Observe that if f is constant, then M(s) = m, which is the polarization tensor given by Beretta et al. [6, 7]. 1 One of the main feature of our result is the following. Unlike the case of the weakly oscillating thin membrane (see [16]), if the quasi -period of the oscillations of the rough layer is fast compared to its thickness, then the layer influence on the steady-state potential may not be approximated by only considering the mean effect of the rough layer. Actually, if we were to consider the mean effect of the roughness, the approximate transmission conditions would be these presented in (13), by replacing f by its average f defined in (7). Observe that our transmission conditions (11) are different since they involve parameters r 1 and r 2 quantifying the roughness of m. More precisely, denote by z the correction, which only takes into account the mean effect of the layer. hen according to (13), z will satisfy (for simplicity, we consider the -periodic case):. (σ k z) = in k, k =, 1, ( ) z + z σ = 1 f u+ on Γ, σ m ν z + σ ν σ z ( ) 1 ν = f(σ σ m ) 2 u θ 2 z =. on Γ, o illustrate this assertion, we conclude the paper by numerical simulations obtained using the mesh generator Gmsh [13] and the finite element library Getfem++ [18]. he computational domain is delimited by the circles of radius 2 and of radius.2 centered in, while 1 is the intersection of with the concentric disk of radius 1. he rough layer is then described by f(y) = sin(y) and we choose β = 1/2. One period of the domain is shown Fig. 2(a). he Dirichlet boundary data is identically 1 on the outer circle and on the inner circle. he conductivities σ, σ 1 and σ m are respectively equal to 1, 1 and.1. he computed coefficients for quantifying the roughness are r 1 = 5.87 and r 2 =.413 (three significant digits are kept). he numerical convergence rates for both the H 1 - and the L 2 -norms in 1 of the three following errors u u, u u β z and u u β z as goes to zero are given Fig. 3 for 6 β = 1/2. he numerical convergence rates with the thickness of the layer are comparable between the H 1 - and the L 2 -norms. 6 he same numerical simulations have been performed for several values of β < 1. All the results are very similar, hence we just show here the case β = 1/2. RR n 6975

23 2 Ciuperca& Perrussel & Poignard (a) One period. (b) Error order. (c) Error order 1. Figure 2: Representation of one period of the domain and the corresponding errors with approximate solutions u and u + β z. = 2π/6. Do not consider the error inside the rough layer because a proper reconstruction of the solution in it is not currently implemented. INRIA

24 Approximate transmission conditions for very rough thin layers 21 Observe that they are also similar to the rates shown in [17, 16] and in [11], respectively for the case of constant thickness and for the periodic roughness case. More precisely they are close to 1 for u u and for u (u+ β z), whereas the convergence rate is close to 2 for u (u + β z). herefore according to these numerical simulations, the convergence of z to z seems to hold strongly in H 1 far from the layer, even if our method does not lead to such result: another analysis should be performed. o conclude, Fig. 4 demonstrates that the convergence rate decreases dramatically for β = 1. his is in accordance with the theory, since the approximate transmission conditions for β = 1 given in [11, 12] are very different from the conditions proved in the present paper. References [1]. Abboud and H. Ammari. Diffraction at a curved grating: M and E cases, homogenization. J. Math. Anal. Appl., 22(3): , [2] Y. Achdou and O. Pironneau. Domain decomposition and wall laws. C. R. Acad. Sci. Paris Sér. I Math., 32(5): , [3] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23(6): , [4] G. Allaire and M. Amar. Boundary layer tails in periodic homogenization. ESAIM Control Optim. Calc. Var., 4: (electronic), [5] H. Ammari and H. Kang. Reconstruction of conductivity inhomogeneities of small diameter via boundary measurements. In Inverse problems and spectral theory, volume 348 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, 24. [6] E. Beretta and E. Francini. Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities. In Inverse problems: theory and applications (Cortona/Pisa, 22), volume 333 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, 23. [7] E. Beretta, E. Francini, and M. S. Vogelius. Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. J. Math. Pures Appl. (9), 82(1): , 23. [8] E. Beretta, A. Mukherjee, and M. S. Vogelius. Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys., 52(4): , 21. [9] G.C. Buscaglia, I.S. Ciuperca, and M. Jai. opological asymptotic expansions for the generalized Poisson problem with small inclusions and applications in lubrication. Inverse Problems, 23(2): , 27. [1] Y. Capdeboscq and M. S. Vogelius. A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. M2AN Math. Model. Numer. Anal., 37(1): , 23. RR n 6975

25 22 Ciuperca& Perrussel & Poignard (a) L 2 error (b) H 1 error Figure 3: Error in the cytoplasm vs β for three approximate solutions. We choose β = 1/2. INRIA

26 Approximate transmission conditions for very rough thin layers 23 Figure 4: L 2 -error in the cytoplasm vs for four approximate solutions. [11] I.S. Ciuperca, M. Jai, and C. Poignard. Approximate transmission conditions through a rough thin layer. he case of the periodic roughness. o appear in European Journal of Applied Mathematics. Research report INRIA RR [12] I.S. Ciuperca, R. Perrussel, and C. Poignard. Influence of a Rough hin Layer on the Steady-state Potential. Research report INRIA RR [13] C. Geuzaine and J. F. Remacle. Gmsh mesh generator. [14] W. Jäger, A. Mikelić, and N. Neuss. Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput., 22(6): (electronic), 2. [15] Dag Lukkassen, Gabriel Nguetseng, and Peter Wall. wo-scale convergence. Int. J. Pure Appl. Math., 2(1):35 86, 22. [16] C. Poignard. Approximate transmission conditions through a weakly oscillating thin layer. Math. Meth. App. Sci., 32: , 29. [17] C. Poignard, P. Dular, R. Perrussel, L. Krähenbühl, L. Nicolas, and M. Schatzman. Approximate conditions replacing thin layer. IEEE rans. on Mag., 44(6): , 28. [18] Y. Renard and J. Pommier. Getfem finite element library. RR n 6975

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