Effective Models for Long Time Wave Propagation in Locally Periodic Media

MATHICSE

Institute of Mathematics School of Basic Sciences

MATHICSE Technical Report Nr. 15.2017

July 2017

Effective models for long time wave propagation in locally periodic media

Assyr Abdulle, Timothé e Pouchon

http://mathicse.epfl.ch Address: EPFL - SB – INSTITUTE of MATHEMATICS – Mathicse (Bâtiment MA) Station 8 - CH-1015 - Lausanne - Switzerland

EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA

ASSYR ABDULLE∗‡ AND TIMOTHEE´ POUCHON†‡

Abstract. A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54, 2016, pp. 1507–1534] is generalized to the case of a locally periodic tensor.

Key words. homogenization, eﬀective equations, wave equation, heterogeneous media, long time behavior, dispersive waves, a priori error analysis, multiscale method

AMS subject classiﬁcations. 35B27, 74Q10, 74Q15, 35L05, (65M60, 65N30)

1. Introduction. The wave equation in heterogeneous media is used to model diverse multiscale applications in engineering such as seismic inversion, medical imaging or the manufacture of composite materials. In such situations, the medium is described by a tensor aε, where ε> 0 denotes the characteristic length of the spatial variation of aε and is assumed to be much smaller than the wavelength of the initial data and the source term (ε ≪ 1). The displacement of the wave uε : [0,T ] × Rd → R is then characterized by the equation

2 ε ε ε Rd ∂t u (t, x) − ∇x · a (x)∇xu (t, x) = f(t, x) in (0,T ] × , (1.1) ε ε where initial conditions for u (0, x) and ∂tu (0, x) are given. Before discretizing (1.1), we truncate the space Rd to a sufficiently large hypercube Ω, so that the waves do not reach the boundary, and impose periodic boundary conditions (Ω is called a pseudoinfinite domain). To approximate (1.1) accurately, standard numerical methods such as the finite element (FE) method or the finite difference (FD) method require a grid that resolves the whole domain at the microscopic scale O(ε). Hence, as T increases (i.e. Ω increases) or as ε → 0, such methods have a prohibitive computational cost. Therefore, more sophisticated numerical methods are needed. The study of multiscale problems such as (1.1) is tied to homogenization theory (see [17, 42, 16, 35, 23, 38]). The general homogenization result for the wave equation in [19] provides the existence of a function 0 ε ∞ 0 u such that the sequence {u }ε>0 converges weakly in L (0,T ; Wper(Ω)) to u as ε → 0 (see below for the definitions of the functional spaces). The homogenized solution u0 is characterized by the homogenized equation

2 0 0 0 ∂t u (t, x) − ∇x · a (x)∇xu (t, x) = f(t, x) in (0,T ] × Ω, (1.2) where the initial conditions are the same as for uε. As the homogenized tensor a0 in (1.2) is obtained as ε the so called G-limit of the sequence {a }ε>0 (see [44, 24]), (1.2) does not depend on the microscopic scale and is thus a good target for numerical methods. However, for a general tensor aε, a0 might not be unique and no formula is available for its computation. Nevertheless, when the medium is locally periodic, i.e., ε x 0 a (x)= a x, ε with y 7→ a(x, y) Y -periodic, such formula exists. Indeed, in this case a (x) can be computed at each x ∈ Ω via the solutions of d cell problems, which are elliptic partial diﬀerential equations (PDEs) in Y (see e.g. [10, 22]). In the past few years, several multiscale methods for the approximation of (1.1) have been developed. The physical origin of (1.1) motivates the choice of an appropriate method. In particular, the problems are divided in two classes, depending whether the medium has, or not, scale separation. Let us ﬁrst mention the methods available if the medium does not have scale separation. We refer to [6] for a detailed review.

∗assyr.abdulle@epfl.ch †timothee.pouchon@epfl.ch ‡ANMC, Section de Mathématiques, Ecole´ Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland 1 2 A. ABDULLE AND T. POUCHON

The methods defined in [39], [34, 33], [40] and [7] rely on multiscale FE spaces that have the same number of degrees of freedom (DOF) as in a coarse FE method. However, the construction of these spaces involves the solutions of global elliptic PDEs at the fine scale, which is computationally expensive and might be prohibitive. To settle this issue, the elliptic PDEs are localized to small patches covering the domain, leading to a process that can be parallelized. Let us then introduce the methods available when the medium has scale separation. In such media, numerical methods can take advantage of the specific structure to reduce the computational cost. To that purpose, the heterogeneous multiscale method (HMM) provides an appropriate framework (see [2]). In the HMM, the effective datum is approximated with a sampling strategy by solving local micro problems and is then used at the macro scale with a chosen numerical method. As the micro scale is resolved only locally in small domains, the cost of the HMM is proportional to the number of DOF at the macro scale. Furthermore, as the micro problems are independent, the sampling procedure can be efficiently parallelized. Two HMMs are available to approximate (1.1). The FD-HMM, defined in [28] and analyzed in [13], relies on a FD method at the macro scale. The effective flux is approximated by solving micro problems in space-time sampling domains of size τ × ηd, where τ, η ≥ ε. The FE-HMM, defined and analyzed in [3], relies on the FE method on a macro mesh to approximate the homogenized solution. The homogenized tensor is approximated at the quadrature points by solving micro problems in spatial sampling domains of size δd, where δ ≥ ε. In the case of a locally periodic tensor, the FD-HMM and the FE-HMM are proved to converge to the homogenized solution u0. When considering large timescales T = O(ε−2), uε develops macroscopic dispersive effects. As the homogenized solution does not describe these effects, new numerical methods are needed for the long time approximation of (1.1). In particular, we look for a new effective equation that captures the dispersion. In the literature, several papers [43, 32, 31, 36, 25, 26, 9, 11, 8] investigated the research of long time effective ε x equations in the case of a uniformly periodic tensor, i.e., a (x) = a ε with a(y) Y -periodic. In particular, a recent result in [8] defines a family of effective equations whose elem ents are proved to approximate uε at large timescales O(ε−2). The family is composed of equations of the form (we use the convention that repeated indices are summed)

2 0 2 2 2 4 2 2 2 −2 ∂t u˜(t, x) − aij ∂ij u˜(t, x)+ ε aijkl∂ijklu˜(t, x) − bij ∂ij ∂t u˜(t, x) = f(t, x) in (0,ε T ] × Ω, (1.3) with the same initial conditions as for uε, where a0 is the homogenized tensor (constant in the uniformly periodic case) and a2, b2 are non-negative tensors that satisfy a given constraint. The two HMMs described above have been adapted to the long time approximation of (1.1). In [29], a modification of the FD-HMM is built to capture the effective flux of an ill-posed effective equation derived in [43]. However, to do so, the space-time sampling strategy requires larger sampling domains as ε → 0. Furthermore, as it is build on an ill-posed model, a regularization step has to be performed. Nevertheless, in one dimension and for uniformly periodic tensors, the method is shown in [12] to capture the effective flux of the ill-posed model. In [5, 4], the FE-HMM was also generalized for long time approximation. The method, called the FE-HMM-L, was analyzed over long time in [9]. In particular, in one dimension and for uniformly periodic tensors, the method is proved to converge to an effective equation of the family (1.3). In this paper, we generalize the family of effective equations from [8] to the case of a locally periodic tensor. This analysis constitutes the first result in the study of long time wave propagation in locally periodic media. The family consists of equations of the form

2 0 1 2 2 −2 ∂t u˜(t, x) − ∂i aij (x)∂j u˜(t, x) + εL u˜(t, x)+ ε L u˜(t, x)= f(t, x) in (0,ε T ] × Ω, (1.4) with the same initial conditions as uε and where the operators L1 and L2 are given as

1 12 10 2 2 2 24 2 22 2 22 20 2 L = −∂i aij (x)∂j · + b ∂t ,L = ∂ij aijkl(x)∂kl · − ∂i bij (x)∂j ∂t · − ∂i aij (x)∂j · + b ∂t . The tensors amn,bmn are defined for all x ∈ Ω via the solutions of local cell problems and are linked by a parameter. The main result of the paper ensures that any effective solution of the family (1.4) satisfies the error estimate

ε ku − u˜kL∞(0,ε−2T ;W ) ≤ Cε, (1.5)

2 where the norm k·kW is defined in (1.8) and is equivalent to the L norm through the Poincaréconstant. As we track the dependency of the estimate on Ω, the result is valid in arbitrarily large hypercubes. Thanks to EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 3 this result, we prove that, in the one-dimensional case, the FE-HMM-L converges to an effective solution in the locally periodic case. In particular, the approximation uH satisfies the error estimate

ε 2 2 ℓ 2 ku − uH kL∞(0,ε−2T ;W ) ≤ C ε + (h/ε ) + H /ε , (1.6) where h is the micro mesh size, H is the macro mesh size, and ℓ is the macro FE degree. Note that, in the last two terms, a factor ε−2 comes from the timescale O(ε−2). We emphasize that thanks to a new elliptic projection, (1.6) can be used in arbitrarily large domain Ω. This result generalizes the long time a priori error analysis of the FE-HMM-L performed in [9]. The paper is organized as follows. First, in Section 2, we present our main result: we deﬁne the family of eﬀective equations (1.4) and state the corresponding error estimate. Then, the derivation of the family and the construction of the adaptation are presented in Section 3 and the proof of the main result is performed in Section 4. In Section 5, we provide long time a priori error estimates for the FE-HMM-L in the locally periodic case. Finally, we illustrate our theoretical results in numerical experiments in Section 6.

Definitions and notation. Let us give some definitions and the notation used in the paper. The derivative with respect to the i-th space variable xi is denoted ∂i and the derivation with respect to any other variable is specified. We denote the quotient space L2(Ω) = L2(Ω)/R and a bracket [v] is used to denote the equivalence class of v ∈ L2(Ω) in L2(Ω). Equipped with the inner product

2 [v], [w] = v − hviΩ, w − hwiΩ = v, w 2 − |Ω|hviΩhwiΩ ∀v, w ∈ L (Ω), L2(Ω) L2(Ω) L (Ω) 2 1 R L (Ω) is a Hilbert space. Furthermore, we denote Wper(Ω) = Hper(Ω)/ and a bold face letter v is used 2 to denote the elements of Wper(Ω). The space Wper(Ω) (resp. L0(Ω)) is composed of the zero mean repre- 2 sentatives of the equivalence classes in Wper(Ω) (resp. L (Ω)). We deﬁne the following norm on Wper(Ω)

kwk = inf k[w ]k 2 + k∇w k 2 ∀w ∈W (Ω), (1.7) W w w w 1 L (Ω) 2 L (Ω) per = 1+ 2 n o wi=[wi]∈Wper(Ω) and the corresponding norm on Wper(Ω)

kwkW = inf kw1kL2(Ω) + k∇w2kL2(Ω) ∀w ∈ Wper(Ω). (1.8) w=w1+w2 n o w1,w2∈Wper(Ω)

We verify that a function w ∈ Wper(Ω) satisfies kwkW = k[w]kW . Furthermore, using the Poincaré– 2 Wirtinger inequality, we verify that k·kW is equivalent to the L norm (CΩ is the Poincaréconstant)

kwkW ≤kwkL2(Ω) ≤ max{1, CΩ}kwkW ∀w ∈ Wper(Ω). (1.9)

We denote Tenn(Rd) the vector space of tensors of order n. In the whole text, we drop the notation of the sum symbol for the dot product between two tensors and use the convention that the repeated indices are summed. The subspace of Tenn(Rd) of symmetric tensors is denoted Symn(Rd), i.e., q ∈ Symn(Rd) satisfies n n Rd n Rd qi1···in = qiσ(1) ···iσ(n) for any permutation σ. Let S : Ten ( ) → Sym ( ) be the symmetrization operator n 1 n n defined as S (q) = qi ···i . In the text, S (q) is denoted as S {qi1···in }. For i1···in n! σ∈Sn σ(1) σ(n) i1···in i1···in 4 d 2 d 4 d q ∈ Ten (R ) andξ, η ∈ Sym P(R ), we denote the product qξ :η = qijkl ξij ηkl. A tensor q ∈ Ten (R ) is major 2 d symmetric if qijkl = qklij for all 1 ≤ i,j,k,l ≤ d and it is positive semidefinite if qξ : ξ ≥ 0 for all ξ ∈ Sym (R ). For a major symmetric tensor q ∈ Ten4(Rd), there exists a bijective map ν : Sym2(Rd) → RN(d), where d+1 2 RN(d) N(d)= 2 , and a matrix M(q) ∈ Sym ( ) such that (see e.g. [8] or [41, Chapter 4]) qξ : η = M(q)ν(ξ) · ν(η) ∀ξ, η ∈ Sym2(Rd). (1.10)

In particular, q is positive (semi)deﬁnite if and only if M(q) is positive (semi)deﬁnite. 4 A. ABDULLE AND T. POUCHON

Settings of the problem. We assume that d ≤ 3 (note that the main result holds for d> 3, provided ε x higher regularity assumption of the tensor). Let a (x) = a x, ε be a d × d symmetric locally periodic tensor, i.e., a(x, y) is Y -periodic in y and Ω-periodic in x, where Ω, Y ∈ Rd are open hypercubes. We assume that Ω is a union of cells of volume ε|Y |. More precisely, letting ℓ ∈ Rd be the period of y 7→ a(x, y) (i.e., Zd l r l r a(x, y + k · ℓ) = a(x, y) for all (x, y) ∈ Ω × Y and k ∈ ), we assume that Ω = (ω1,ω1) ×···× (ωd,ωd) satisﬁes r l ωi − ωi ∈ N>0 ∀i =1, . . . , d. (1.11) εℓi

x This assumption ensures that for any Y -periodic function γ, the map x 7→ γ ε is Ω-periodic (γ is extended to Rd by periodicity). For T ε = ε−2T , we consider the wave equation: uε : [0,Tε] × Ω → R such that

2 ε x ε ε ∂t u (t, x) − ∇x · a x, ε ∇xu (t, x) = f(t, x) in (0,T ] × Ω, x 7→ uε(t, x) Ω-periodic in [0,T ε], (1.12) ε 0 ε 1 u (0, x)= g (x), ∂tu (0, x)= g (x) in Ω, where g0,g1 are given initial conditions and f is a source. The tensor a(x, y) is assumed to be uniformly elliptic and bounded, i.e. there exists λ, Λ > 0 such that

λ|ξ|2 ≤ a(x, y)ξ · ξ ≤ Λ|ξ|2 ∀ξ ∈ Rd for a.e. (x, y) ∈ Ω × Y. (1.13)

0 1 2 2 ε 2 The well-posedness of (1.12) is proved in [37, 30]. If g ∈ Wper(Ω), g ∈ L0(Ω), f ∈ L (0,T ;L0(Ω)), ε ∞ ε ε ∞ ε 2 2 ε then there exists a unique weak solution u ∈ L (0,T ; Wper(Ω)) with ∂tu ∈ L (0,T ;L0(Ω)) and ∂t u ∈ 2 ε ∗ L (0,T ; Wper(Ω)). 2. Main result: definition of the family of effective equations and a priori error estimate. In this section we present the main result of the paper. We define the family of effective equations and state the a priori error estimate. Let us first define the operators involved in the definition of the family of effective equations. For all d 0 d 1 d x ∈ Ω, let {χi(x)}i=1, {θij (x)}ij=1, {θi (x)}i=1 ⊂ Wper(Y ) be the zero mean solutions of the cell problems

a(x)∇yχi(x), ∇yw Y = − a(x)ei, ∇yw Y , (2.1a) 0 0 a(x)∇yθij (x), ∇yw Y = − a(x)eiχj (x), ∇yw Y + a(x)(∇yχj(x)+ ej ) − a (x)ej ,eiw Y , (2.1b) 1 0 a(x)∇yθi (x), ∇yw Y = − a(x)∇xχi(x), ∇yw Y + ∇x · a(x)(∇yχi(x)+ ei) − ∇x · a (x)ei, w Y , (2.1c) 0 for all test functions w ∈ Wper(Y ), where a (x) is the homogenized tensor deﬁned by

0 T aij (x)= ei a(x)(∇yχj (x)+ ej ) Y . (2.2)

We deﬁne the diﬀerential operator

1 12 10 2 L = −∂i a¯ij (x)∂j · + b ∂t , (2.3) based on the following tensors

13 pijk(x)= a(x)(∇yχk(x)+ ek) · ej χi(x) Y , 12 qij (x)= a(x)(∇yχj (x)+ ej ) · ∇xχi(x) Y , 12 2 13 13 13 12 aˇij (x)= Sij − ∂mpmij (x)+ ∂mpjim(x) − ∂mpimj (x)+2qij (x) , n o 12 10 λmin(ˇa (x)) b = max − 0 , x∈Ω λmin(a (x)) + (2.4) 12 12 10 0 a¯ij (x)=ˇaij (x)+ b aij (x), where we denoted {·}+ = max{0, ·}. Furthermore, we deﬁne the diﬀerential operator

2 2 24 2 22 2 22 20 2 L = ∂ij a¯ijkl(x)∂kl · − ∂i bij (x)∂j ∂t · − ∂i a¯ij (x)∂j · + b ∂t , (2.5) EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 5 deﬁned upon the following tensors and functions

24 2,2 0 0 aˇijkl(x)= Sij,kl a(x)χi(x)ej · χl(x)ek Y − a(x)∇yθij (x) · ∇yθkl(x) Y , n o 24 24 0 2,2 0 0 A (x)= M aˇ (x) , A (x)= M Sij,kl ajk(x)ail(x) , 24 ∗ λmin(A (x)) δ ≥ δ = max − 0 , x∈Ω λmin(A (x)) + 24 24 2,2 0 0 a¯ijkl(x)=ˇaijkl(x)+ δS ajk(x)ail(x) , ij,kl (2.6) 22 0 bij (x)= χi(x)χj (x) Y + δaij (x),

2,2 2 2 where Sij,kl{·} = Sij {Skl{·}} and M(·) is given in (1.10) and

23 0 1 pijk(x)= a(x)ej χi(x) · ∇xχk(x) Y − a(x)∇yθji(x) · ∇yθk(x) Y , 22 1 1 pij (x)= a(x)∇xχj (x) · ∇xχi(x) Y − a(x)∇yθi (x) · ∇yθj (x) Y , (2.7) 22 2 23 23 23 22 aˇij (x)= Sij − ∂mpmij (x)+ ∂mpjim(x) − ∂mpimj (x)+ pij (x) n o 10 12 0 0 0 0 + b aˇij (x)+ δ∂nami(x)∂manj (x) − δ∂m amn(x)∂naij (x) , 22 20 λmin(ˇa (x)) b = max − 0 , x∈Ω λmin(a (x)) + (2.8) 22 22 20 0 a¯ij (x)=ˇaij (x)+ b aij (x). Observe that the tensors of L2 are parametrized by δ ≥ δ∗. Let thenu ˜ : [0,T ε] × Ω → R be the solution of

2 0 1 2 2 ε ∂t u˜(t, x) − ∂i aij (x)∂j u˜(t, x) + εL u˜(t, x)+ ε L u˜(t, x)= f(t, x) in (0,T ] × Ω, x 7→ u˜(t, x) Ω-periodic in [0,T ε], (2.9) 0 1 u˜(0, x)= g (x), ∂tu˜(0, x)= g (x) in Ω, where the initial conditions g0,g1 and the source f are the same as in the equation for uε (1.12). It is known that the homogenized tensor a0 is symmetric, uniformly elliptic, and bounded. Furthermore, note that by definition,a ¯12,a ¯22,b22 are symmetric and positive semidefinite, b10,b20 are non-negative, anda ¯24 is major 24 24 symmetric (i.e.a ¯ijkl =a ¯klij ) and positive semidefinite (see [8, Lemma 4.2] for a similar result). We verify that if the tensor a(x, y) satisfies a ∈C2(Ω;L¯ ∞(Y )), then a0, a¯24,b22 ∈C2(Ω),¯ a ¯12 ∈C1(Ω)¯ anda ¯22 ∈C0(Ω)¯ (see (4.2)). If, in addition, the data satisfy the regularity

0 2 1 2 1 2 ε 2 g ∈ Wper(Ω) ∩ H (Ω), g ∈ L0(Ω) ∩ H (Ω), f ∈ L (0,T ;L0(Ω)), then there exists a unique weak solution of (2.9) (see e.g. [41, Chapter 2]). Definition 2.1. We define the family of effective equations E as the set of equations (2.9), where a0 is the homogenized tensor defined in (2.2) and L1, L2 are defined in (2.3) and (2.5) for some parameter δ ≥ δ∗. Remark 2.2. For uniformly periodic tensors, i.e. a(x, y) = a(y) ∀x ∈ Ω, the family E simplifies to a parametrized subset of the family defined in [8]. Indeed, in that case we verify that L1 = 0 and 2 24 4 22 2 2 24 22 L =a ¯ijkl∂ijkl − b ∂ij ∂t , wherea ¯ ,b are constant and satisfy the constraint characterizing the family from [8]. Our main result is the following theorem. Theorem 2.3. Assume that the tensor a(x, y) satisfies a ∈C1(Ω;¯ W2,∞(Y )) ∩C2(Ω;¯ W1,∞(Y )) ∩C4(Ω;L¯ ∞(Y )). Furthermore, assume that the solution u˜ of (2.9), the initial conditions, and the right hand side satisfy the regularity

∞ ε 5 ∞ ε 4 2 ∞ ε 3 u˜ ∈ L (0,T ; H (Ω)), ∂tu˜ ∈ L (0,T ; H (Ω)), ∂t u˜ ∈ L (0,T ; H (Ω)), g0 ∈ H4(Ω), g1 ∈ H4(Ω), f ∈ L2(0,T ε; H2(Ω)). 6 A. ABDULLE AND T. POUCHON

Then the following estimate holds

ε 1 0 ku − u˜kL∞(0,T ε;W ) ≤ Cε kg kH4(Ω) + kg kH4(Ω) + kfkL1(0,T ε;H2(Ω)) (2.10) 5 ∞ 2 ∞ + k=1 |u˜|L (0,T ε;Hk(Ω)) + k∂t u˜kL (0,T ε;H3(Ω)) , P ˜ ˜ where C = C kakC1(Ω;W¯ 2,∞Y )) + kakC2(Ω;W¯ 1,∞(Y )) + kakC4(Ω;L¯ ∞(Y )) and C depends only on T , λ, Y , and δ, (the norm k·kW is defined in (1.8)). We emphasize that the constant C˜ is independent of the domain Ω. Hence, if the different norms of the ε data involved in the estimate are of order O(1), (2.10) ensures that ku −u˜kL∞(0,T ε;W ) = O(ε). In particular, if the data have a spatial support of order O(1), a reasonable spatial variation, and if f has a temporal support of order O(1), thenu ˜ describes well uε over the long time interval [0,T ε]. 3. Derivations of the adaptation operator and effective equations. In this section, we proceed with the asymptotic expansion and construct the adaptation operator required in the proof of Theorem 2.3, performed in the next section. As we will see, this construction is connected to the operators involved in the family of effective equations defined in Definition 2.1. The main result of this section is the following theorem. Theorem 3.1. Let L1 and L2 be defined in (2.3) and (2.5), respectively. Then there exists an adaptation of the form

ε 1 x 2 2 x 3 3 x 4 4 x B u˜(t, x)=˜u(t, x)+ εu t, x, ε + ε u t, x, ε + ε u t, x, ε + ε u t, x, ε + ϕ(t, x), (3.1) such that x 7→ Bεu˜(t, x) is Ω-periodic and

ε ε ε ε (u −B u˜)(0) = O(ε), ∂t(u −B u˜)(0) = O(ε), (3.2a) 2 ε ε ε 3 ε (∂t + A )(u −B u˜)(t)= O(ε ) for a.e. t ∈ [0,T ], (3.2b)

ε x where we denoted A = −∇x · a x, ε )∇x · . Thanks to (3.2), under sufficient regularity assumptions, the adaptation can be proved to satisfy ε ε k[u −B u˜]kL∞(0,T ε;W) = O(ε) and Theorem 2.3 is obtained with the triangle inequality (4.1) (this is done rigorously in the next section). Let us note that the accuracy required on the adaptation in (3.2b) is dictated by the order of the timescale T ε = O(ε−2) (see [8] or [41, Chapter 4]). In the rest of the section, we proceed with the construction of the adaptation Bεu˜ and of the effective equations. In particular, we need to define the functions uk and ϕ in (3.1) so that (3.2) holds. We will see that the definitions of L1 and L2 enable the definitions of u3 to u4, respectively. Before entering into technical details, let us present a plan of the construction. First, we formulate the ansatz that an effective equation has the form (2.9), where a0(x) is the homogeneous tensor (defined in (2.2)) and L1,L2 are ε-independent differential operators to be defined. To emphasize that L1,L2 are unknown at ˜1 ˜2 ε 2 ε ε ε this point, let us denote them as L and L . We then expand R = (∂t + A )(u −B u˜)(t) with the aim to attain the accuracy (3.2b). Canceling one after another the terms of Rε of order O(ε−1) to O(ε2), each uk takes the form

k uk(t,x,y)= ck,ℓ x, x ∂k−ℓ+1 u˜(t, x), i1··ik−ℓ+1 ε i1··ik−ℓ+1 Xℓ=1 where the corrector ck,ℓ (x, ·) solves a cell problem in Y (i.e., an elliptic PDE with periodic boundary i1··ik−ℓ+1 conditions). The well-posedness of these cell problems imposes quantitative constraints on L˜1 and L˜2. We then design L˜1 and L˜2 so that these constraints are satisfied and (2.9) is well-posed. In what follows, we require the correctors ck,ℓ (x, ·) to have zero mean. While this is a priori not necessary, it is a natural i1··ik−ℓ+1 choice and simplifies the computations. Let us now present the technical details of the derivation. We introduce the differential operators

Ayy = −∇y · a(x, y)∇y · , Axy = −∇y · a(x, y)∇x · − ∇x · a(x, y)∇y · , Axx = −∇x · a(x, y)∇x · . EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 7

ε x −2 −1 x For a suﬃciently regular function ψ(x, y), we verify that A ψ x, ε = ε Ayy + ε Axy + Axx ψ x, ε . Hence, using (1.12), (2.9) and (3.1), we obtain the development

ε 2 ε ε ε 2 ε ε ε R = (∂t + A )(B u˜ − u )(t, x)= ∂t B u˜(t, x)+ A B u˜(t, x) − f(t, x) −1 1 = ε Ayyu + Axyu˜ 0 2 1 0 + ε Ayyu + Axyu + Axxu˜ + ∂i(aij ∂j u˜) (3.3) 1 2 1 3 2 1 ˜1 + ε ∂t u + Ayyu + Axyu + Axxu − L u˜ 2 2 2 4 3 2 ˜2 + ε ∂t u + Ayyu + Axyu + Axxu − L u˜ 2 ε 3 + (∂t + A )ϕ + O(ε ),

i x 1 4 where the u are evaluated at t,x,y = ε . We then look for u ,...,u and ϕ such that the terms of order O(ε−1) to O(ε2) in (3.3) vanish. Note that the uk are set to cancel the terms containingu ˜ and ϕ are set to cancel the terms containing f (that will appear). 3.1. Canceling the ε−1,ε0 and ε terms and derivation of the constraints defining L˜1. To cancel the term of order O(ε−1) in (3.3), it is sufficient to define

1 u (t,x,y)= χi(x, y)∂iu˜(t, x), (3.4) where, for all x ∈ Ω and 1 ≤ i ≤ d, χi(x) ∈ Wper(Y ) solves the cell problem

−1 ε : a(x)∇yχi(x), ∇yw Y = − a(x)ei, ∇yw Y , (3.5) for all test functions w ∈ Wper(Y ). To prove the well-posedness of (3.5), we apply Lax–Milgram theorem. ∗ In particular, we must verify that the right hand side belongs to Wper(Ω). To do so, we need the follow- 1 ∗ ing characterization (consequence of Riesz representation theorem): a functional F ∈ [Hper(Y )] , given by 0 1 0 1 1 2 ∗ 0 F, w = f , w Y + fk , ∂kw Y , for some f ,f1 ,...,fd ∈ L (Y ), belongs to Wper(Y ) if and only if f is zero mean, or equivalently

0 f , 1 Y =0. (3.6) ∗ Using the characterization (3.6), we verify that the right hand side of (3.5) belongs to Wper(Y ) and the equation is thus well-posed in Wper(Y ). The equation obtained by canceling the term of order O(1) in (3.3) reads now

2 T 0 2 −∇y · (a∇yu )= ∇y · (eiχj )+ ei a(∇yχj + ej) − aij ∂ij u˜ 0 + ∇y · (∇xχi)+ ∇x · a(∇yχi + ei) − ∇x · (a ei) ∂iu.˜ Compared to the uniformly periodic case in [8], we observe that a supplementary term coming from the variation x 7→ a(x, y) appears in this equation. To satisfy this equality, it is suﬃcient to deﬁne

2 0 2 1 u (t,x,y)= θij (x, y)∂ij u˜(t, x)+ θi (x, y)∂iu˜(t, x), (3.7)

0 1 where, for all x ∈ Ω and 1 ≤ i, j ≤ d, θij (x),θi (x) ∈ Wper(Y ) satisfy

ε0 : 0 0 a(x)∇yθij (x), ∇yw Y = − a(x)eiχj (x), ∇yw Y + a(x)(∇yχj (x)+ ej ) − a (x)ej ,eiw Y , (3.8a) 1 0 a(x)∇yθi (x), ∇yw Y = − a(x)∇xχi(x), ∇yw Y + ∇x · a(x)(∇yχi(x)+ ei) − ∇x · (a (x)ei), w Y , (3.8b) ∗ for all test functions w ∈ Wper(Y ). To verify that the right hand sides of these PDEs belong to Wper(Y ), we check that they satisfy (3.6). Thanks to the deﬁnition of the homogenized tensor a0 in (2.2), the right hand side of (3.8a) satisﬁes (3.6) for all x ∈ Ω. Furthermore, we have

0 T 0 ∇x · a(∇yχi + ei) − ∇x · (a ei), 1 Y = |Y |∂m ema(∇yχi + ei) Y − ami =0, 8 A. ABDULLE AND T. POUCHON and the right hand side of (3.8b) also satisfies (3.6). Hence, both cell problems in (3.8) are well-posed. At this point, we have defined an adaptation such that Rε = O(ε), which would be sufficient for a timescale O(1). As the timescale is of order O(ε−2), we need the accuracy Rε = O(ε3) in (3.2b), and we thus continue to cancel the higher order terms in (3.3). We begin with the terms containingu ˜. Taking into account the definitions of u1 and u2 and the effective equation (2.9), we have 2 1 2 0 ˜1 2 ∂t u = χi∂i∂t u˜ = χi∂if + χi∂im(amn∂nu˜) − εχi∂iL u˜ + O(ε ), 2 2 0 2 2 1 2 0 2 1 0 3 0 1 0 ∂t u = θij ∂ij ∂t u˜ + θi ∂i∂t u˜ = θij ∂ij f + θi ∂if + θij ∂ijm(amn∂nu˜)+ θk∂km(amn∂nu˜)+ O(ε). Plugging these equalities in (3.3), we obtain ε 3 2 1 2 0 ˜1 R = ε Ayyu + Axyu + Axxu + χi∂im(amn∂nu˜) − L u˜ 2 4 3 2 0 3 0 1 2 0 ˜1 ˜2 + ε Ayyu + Axyu + Axxu + θij ∂ijm(amn∂nu˜)+ θi ∂im(amn∂nu˜) − χi∂iL u˜ − L u˜ (3.9) 2 ε 2 0 2 1 3 +(∂t + A )ϕ+ εχi∂if + ε (θij ∂ij f + θi ∂if)+ O(ε ). We are now looking for u3 such that the O(ε) order term in (3.9) cancels. We thus define

3 0 3 1 2 2 u (t,x,y)= κijk(x, y)∂ijk u˜(t, x)+ κij (x, y)∂ij u˜(t, x)+ κi (x, y)∂iu˜(t, x), (3.10)

0 1 2 ˜1 where κijk(x),κij (x) and κi (x) are solutions of cell problems to be defined. We now need to design L such ˜1 13 3 12 2 11 that these cell problems are well-posed. The first idea is to set L = aijk(x)∂ijk − aij (x)∂ij + ai (x)∂i and to define the tensors a13,a12,a11 using the constraints imposed by the solvability of the cell problems. However, 13 3 we also have to ensure the well-posedness of the effective equation (2.9). We will see that aijk(x)∂ijk = 0. 12 2 0 12 Nevertheless, for the operator −εaij (x)∂ij not to deteriorate the ellipticity of −∂i(aij ∂j ·), a has to be non- negative. This condition can not be ensured in general by the tensor involved by the obtained constraint. 10 2 ˜1 We thus apply a Boussinesq trick: adding the term b ∂t in L , we observe that if we formally substitute 2 0 ˜1 1 ∂t u˜ = f + ∂i(aij ∂j u˜) in L u˜, the constraint imposed by the well-posedness of the cell problem for κij applies 12 10 0 0 10 12 on aij − b aij . As a is positive definite, note that we can then find non-negative b ,a that satisfy the constraint. Let then ˜1 13 3 12 2 11 10 2 L = aijk(x)∂ijk − aij (x)∂ij + ai (x)∂i + b (x)∂t . (3.11) Using the effective equation, we obtain ˜1 ˜1,x 10 0 10 10 ˜1 2 L u˜ = L u˜ + b ∂m(amn∂nu˜)+ b f + εb L u˜ + O(ε ), (3.12) ˜1,x ˜1 10 2 ˜1 where we denoted L = L − b (x)∂t , the spatial part of L . Hence, we rewrite (3.9) as ε 3 2 1 2 0 ˜1,x 10 0 R = ε Ayyu + Axyu + Axxu + χi∂im(amn∂nu˜) − L u˜ − b ∂m(amn∂nu˜) 2 4 3 2 0 3 0 1 2 0 + ε Ayyu + Axyu + Axxu + θij ∂ijm(amn∂nu˜)+ θi ∂im(amn∂nu˜) (3.13) 1 10 1 2 − χi∂i(L˜ u˜)+ b L˜ u˜ − L˜ u˜ 2 ε 2 0 2 1 10 3 +(∂t + A )ϕ+ εχi∂if + ε (θij ∂ij f + θi ∂if) − εb f + O(ε ). Recalling the definition of u3 in (3.10), the cancellation of the O(ε) order term in (3.13) leads to the following 0 1 2 cell problems: for all x ∈ Ω and 1 ≤ i, j, k ≤ d, κijk(x),κij (x),κi (x) ∈ Wper(Y ) satisfy (for readability we do not specify the evaluation in x)

ε1 : 0 0 0 0 13 a∇yκijk, ∇yw Y = − aeiθjk, ∇yw Y + a(∇yθjk + ejχk),eiw Y − aij χk, w Y + aijk, w Y , (3.14a) 1 0 1 0 a∇yκij , ∇yw = − a(∇xθij + eiθj ), ∇yw + ∇x · a(∇yθij + eiχj ), w Y Y Y (3.14b) 1 0 0 12 10 0 + a(∇yθj + ∇xχj ),eiw Y − χi∂mamj + χm∂maij , w Y − aij − b aij , w Y , 2 1 1 2 0 a∇yκi , ∇yw = − a∇xθi , ∇yw + ∇x · a(∇yθi + ∇xχi), w − χm∂mnani, w Y Y Y Y (3.14c) 10 0 11 + b ∂mami + ai , w Y , EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 9 for all test functions w ∈ Wper(Y ). These cell problems are well-posed if their right hand sides satisfy (3.6). ˜1 This is the case if and only if the tensors of L satisfy the following constraints (recall that χk(x) Y = 0): 13 0 |Y |aijk = − a(∇yθjk + ej χk),ei Y , (3.15a) 12 10 0 0 1 |Y |(aij − b aij )= ∇x · a(∇yθij + eiχj ), 1 Y + a(∇yθj + ∇xχj ),ei Y , (3.15b) 11 1 10 0 |Y |ai = − ∇x · a(θi + ∇xχi), 1 Y − |Y |b ∂mami. (3.15c) We emphasize that the constraints (3.15) must hold for each x ∈ Ω. These expressions are simpliﬁed in the following lemma. Lemma 3.2. The constraints on a13,a12,b10 and a11 deﬁned in (3.15) can be rewritten for all x ∈ Ω as 13 13 13 13 aijk (x) = (pijk − pkji)(x), pijk = a(∇yχk + ek) · ej χi Y , (3.16a)

12 10 0 13 12 12 1 (aij − b aij )(x)= −∂mamij (x)+ pij (x), pij = a(∇yθj + ∇xχj ) · ei Y , (3.16b)

11 12 10 0 ai (x)= −∂mpmi(x) − b ∂mami(x). (3.16c) Furthermore, p12(x) satisﬁes 12 13 12 12 12 pij (x)= −∂mpimj (x)+ qij (x)+ qji (x), qij = a(∇yχj + ej ) · ∇xχi Y . (3.16d)

Proof. Let us denote (·, ·)Y as (·, ·) and h·iY as h·i. We first prove (3.16a). Using (3.5) with the test 0 function w = θjk and (3.8a) with w = χi, we have 0 0 − a(∇yθjk + ejχk),ei = a∇yθjk, ∇yχi − aej χk,ei = − aej χk, ∇yχi + ei + a(∇yχk + ek),ej χi , which, thanks to the symmetry of a(x, y) proves (3.16a). Let us now prove (3.16b). Thanks to (3.15a), the first term of (3.15b) is 0 0 13 ∇x · a(∇yθij + eiχj ), 1 = ∂m a(∇yθij + eiχj ),em = −|Y |∂mamij , 11 1 and thus (3.15b) can be rewritten as (3.16b). To rewrite ai as in (3.16c), we simply note that − ∇x ·a(∇yθi + 12 1 ∇xχi), 1 = −|Y |∂mpmi. Finally, let us prove (3.16d). Using (3.5) with the test function w = θj and (3.8b) with w = χi, we have 1 1 a(∇yθj + ∇xχj ),ei = − a∇yθj , ∇yχi + a∇xχj ,ei = a∇xχj , ∇yχi + ei − ∇x · a(∇yχj + ej ),χi . Furthermore, the last term satisfies 13 12 − ∇x · a(∇yχj + ej ),χi = −∂m a(∇yχj + ej ),emχi + a(∇yχj + ej ), ∇xχi = |Y |(−∂mpimj + qij ). Combining the two last equalities gives (3.16d) and the proof of the lemma is complete. In the following proposition, we verify that the two operators L˜1 and L1 coincide. Proposition 3.3. Let a¯12 and b10 be the tensors defined in (2.4) and assume that a¯12 ∈ C1(Ω)¯ . Let also L˜1 and L1 be the operators defined in (3.11) and (2.3), respectively. Then L˜1v = L1v for any v ∈ ∞ ε 3 2 ∞ ε 2 L (0,T ; H (Ω)) with ∂t v ∈ L (0,T ;L (Ω)). 3 13 13 3 Proof. First, note that thanks to (3.16a), we have Sijk{aijk} = 0 and thus aijk∂ijkv = 0. Furthermore, 2 12 12 thanks to (3.16a), (3.16b), and (3.16d), we verify that Sij {aij } =¯aij . Hence, we have ˜1 10 2 2 12 2 11 12 11 2 12 L v − b ∂t v = −Sij {aij }∂ij v + ai ∂iv = −∂i a¯ij ∂j v + ai + ∂m(Smi{ami}) ∂iv. (3.17) 11 2 12 10 We claim that ai + ∂m(Smi{ami}) = 0. To prove it, note that as b is constant, using (3.16b) and (3.16c), we have 11 2 12 1 12 12 1 2 13 13 ai + ∂m(Smi{ami})= 2 ∂m pim − pmi − 2 ∂mn anmi + anim . Using then (3.16a) and (3.16d), we verify that 11 2 12 1 2 13 13 13 13 13 13 ai + ∂m(Smi{ami})= 2 ∂mn − pinm + pmni − pnmi + pimn − pnim + pmin =0, and the claim is proved. Combined with (3.17), the claim concludes the proof of the lemma. 10 A. ABDULLE AND T. POUCHON

3.2. Canceling the ε2 term and derivation of the constraints deﬁning L˜2. We now come back to the asymptotic expansion. The next step is to cancel the O(ε2) order term containingu ˜ in (3.13). Following the same reasoning as for u3, we deﬁne u4 as

4 0 4 1 3 2 2 3 u (t,x,y)= ρijkl(x, y)∂ijkl u˜(t, x)+ ρijk(x, y)∂ijku˜(t, x)+ ρij (x, y)∂ij u˜(t, x)+ ρi (x, y)∂iu˜(t, x), (3.18)

0 1 2 3 ˜2 2 24 2 23 3 22 2 for some ρ ,ρ ,ρ ,ρ to be defined. The ansatz on the form of L could be ∂ij (aijkl∂kl)+ aijk∂ijk − aij ∂ij + 21 ˜1 ai ∂i. However, as for L , this choice does not allow the well-posedness of the effective equation (2.9). We ˜1 20 2 ˜2 thus apply Boussinesq tricks. First, similarly as for L , we add the operator b ∂t in L in order to obtain a 22 20 0 constraint on the difference aij − b aij . Second, inspired by the uniformly periodic case in [8], we add the 22 2 24 0 22 term −∂i(bij ∂j ∂t ·) in order to obtain a constraint on aijkl − ajkbil . Finally, for the operator of order 3, we 23 23 3 will see that we can find a tensor a that satisfies the constraint and aijk∂ijk = 0. We thus define ˜2 2 24 2 22 2 23 3 22 2 21 20 2 L = ∂ij (aijkl(x)∂kl·) − ∂i(bij (x)∂j ∂t ·)+ aijk(x)∂ijk − aij (x)∂ij + ai (x)∂i + b (x)∂t , (3.19) and, using (2.9), we verify that ˜2 ˜2,x 22 2 0 20 0 22 20 L u˜ = L u˜ − ∂i(bij ∂jk(akl∂lu˜)) + b ∂m(amn∂nu˜) − ∂i(bij ∂j f)+ b f + O(ε), (3.20)

˜2,x ˜2 22 2 20 2 ˜2 where L = L + ∂i(bij ∂j ∂t ·) − b ∂t is the spatial part of L . Let us rewrite the following terms of (3.13) taking into account the deﬁnition of L˜1 and using (2.9): ˜1 ˜1,x 10 0 10 χi∂i(L u˜)= χi∂i(L u˜)+ χi∂i(b ∂m(amn∂nu˜)) + χi∂i(b f)+ O(ε), 10 ˜1 10 ˜1,x 10 2 0 10 2 b L u˜ = b L u˜ + (b ) ∂m(amn∂nu˜) + (b ) f + O(ε). Therefore, using the deﬁnitions of the uk, and (3.20), we rewrite the O(ε2) order term in (3.13) and obtain ε 2 4 3 2 0 3 0 1 2 0 ˜1,x R = ε Ayyu + Axyu + Axxu + θij ∂ijm(amn∂nu˜)+ θi ∂im(amn∂nu˜) − χi∂i(L u˜) 10 0 10 ˜1,x 10 2 0 + χi∂i(b ∂m(amn∂nu˜)) − b L u˜ + (b ) ∂m(amn∂nu˜) ˜2,x 22 2 0 20 0 − L u˜ + ∂i(bij ∂jm(amn∂nu˜)) − b ∂m(amn∂nu˜) (3.21) 10 2 0 2 1 10 10 2 22 20 + ε(χi∂if − b f)+ ε θij ∂ij f + θi ∂if − χi∂i(b f) + (b ) f + ∂i(bij ∂j f) − b f 2 ε 3 +(∂t + A )ϕ+ O(ε ).

0 1 2 We thus obtain the following cell problems: for x ∈ Ω and 1 ≤ i,j,k,l ≤ d, ρijkl(x), ρijk(x), ρij (x), 3 ρi (x) ∈ Wper(Y ) satisfy ε2 : 0 0 0 0 a∇yρijkl, ∇yw = − aeiκjkl, ∇yw + a(∇yκjkl + ej θkl),eiw Y Y Y (3.22a) 13 0 0 24 0 22 + ajklχi − aij θkl, w Y + aijkl − ajkbil , w Y , 1 1 0 0 0 a∇yρijk, ∇yw Y = − a(eiκjk + ∇xκijk, ∇yw Y + ∇x · a(∇yκijk + eiθjk), w Y 1 0 1 + a(∇yκjk + ∇xθjk + ej θk),eiw Y 13 10 0 12 0 0 0 0 0 0 0 1 + χm∂maijk + (b aij − aij )χk − θij ∂mamk − θmi∂majk − θim∂majk − aij θk, w Y 24 24 22 0 22 0 22 0 10 13 23 + ∂m(aimjk + amijk − bmkaij ) − bim∂majk − bij ∂mamk − b aijk + aijk, w Y , (3.22b) 2 2 1 1 0 1 a∇yρij , ∇yw Y = − a(eiκj + ∇xκij ), ∇yw + ∇x · a(∇yκij + ∇xθij + eiθj ), w Y 2 1 + a(∇yκj + ∇xθj ),eiw Y 10 0 12 10 0 11 + χm∂m(b aij − aij )+ b χi∂mamj + ai χj (3.22c) 1 0 1 0 0 0 2 0 0 2 0 − θi ∂mamj − θm∂maij − (θim + θmi)∂mnanj − θmn∂mnaij , w Y 2 24 22 0 22 0 22 2 0 + ∂mnamnij − ∂m(bmi∂nanj ) − ∂m(bmn∂naij ) − bim∂mnanj 10 12 10 0 22 20 0 + b (aij − b aij ) − (aij − b aij ), w Y , EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 11

3 2 2 1 a∇yρi , ∇yw Y = − a∇xκi , ∇yw Y + ∇x · a(∇yκi + ∇xθi ), w Y 11 10 0 0 3 0 1 2 0 + χm∂mai + χm∂m(b ∂nani) − θmn∂mnpapi − θm∂mnani, w Y (3.22d) 20 0 10 10 0 11 22 2 0 21 + b ∂mami − b (b ∂mami + ai ) − ∂m(bmn∂npapi)+ ai , w Y , for all test functions w ∈ Wper(Y ). The cell problems (3.22) are well-posed if their right hand sides satisfy ˜2 0 1 (3.6). This is the case if and only if the tensors of L satisfy the following constraints (χi(x),θij (x) and θi (x) have zero mean): 24 0 22 0 0 |Y |(aijkl − ajkbil )= − a(∇yκjkl + ej θkl),ei Y , (3.23a) 23 0 0 1 0 1 |Y |aijk = − ∇x · a(∇yκijk + eiθjk), 1 − a(∇yκjk + ∇xθjk + ej θk),ei Y Y (3.23b) 22 0 22 0 10 13 22 0 24 24 + |Y | bim∂majk + bij ∂mamk + b aijk + ∂m(bmkaij − aimjk − amijk) , 22 20 0 1 0 1 2 1 |Y |(aij − b aij )= ∇x · a(∇yκij + ∇xθij + eiθj ), 1 Y + a(∇yκj + ∇xθj ),ei Y 2 24 22 0 22 0 22 2 0 + |Y | ∂mnamnij − ∂m(bmi∂nanj ) − ∂m(bmn∂naij ) − bim∂mnanj (3.23c) 10 12 10 0 + b (aij − b aij ) , 21 2 1 10 10 0 11 20 0 22 2 0 |Y |ai = − ∇x · a(∇yκi + ∇xθi ), 1 Y + |Y | b (b ∂mami + ai ) − b ∂mami + ∂m(bmn∂npapi) , (3.23d) for all x ∈ Ω. These expressions are simpliﬁed in the following Lemma (we refer to [41, Lemma 6.2.8] for the proof). Lemma 3.4. 22 24 22 23 22 20 Denote Rij (x) = bij (x) − χj (x)χi(x) Y . Then the constraints on a , b , a , a , b and a21 given in (3.23) can be rewritten as 24 0 0 0 aijkl = ajkχlχi Y − a∇yθkl · ∇yθji Y + ajkRil, (3.24a)

23 23 23 10 13 0 0 0 aijk = pijk − pkji + b aijk − ∂m(amj Rik)+ ∂mamkRij + ∂majkRmi, (3.24b)

23 0 1 pijk = aej χi · ∇xχk Y − a∇yθji · ∇yθk Y , (3.24c)

22 20 0 23 23 23 22 10 12 10 0 aij − b aij = ∂m(pjim − pmij − pimj )+ pij + b (aij − b aij ) 2 0 0 0 2 0 (3.24d) + ∂mn(aniRmj ) − ∂m(∂nanj Rmi) − ∂m(∂naij Rmn) − ∂mnanj Rim,

22 1 1 pij = a∇xχj · ∇xχi Y − a∇yθi · ∇yθj Y , (3.24e)

21 2 23 22 10 10 0 11 20 0 2 0 ai = ∂mnpmni − ∂mpmi + b (b ∂mami + ai ) − b ∂mami + ∂m(∂npapiRmn). (3.24f)

We then verify that the two operators L˜2 and L2 coincide. Proposition 3.5. Let a¯24, b22, a¯22 be the tensors defined in (2.6) and (2.8) and assume that a¯24 ∈C2(Ω)¯ and b22, a¯12 ∈C1(Ω)¯ . Let also L2 be the operator defined in (2.5) and L˜2 be the operator defined in (3.19) with 0 R ˜2 2 ∞ ε 4 the tensors given in (3.24) where Rij = δaij for some δ ∈ . Then L v = L v for any v ∈ L (0,T ; H (Ω)) 2 ∞ ε 2 with ∂t v ∈ L (0,T ; H (Ω)). 0 3 23 Proof. First, inserting Rij = δaij in (3.24d) and using (3.16a), we verify that Sijk{aijk} = 0 and 23 3 2 22 22 thus aijk∂ijk v = 0. Second, using (3.24d), (2.4), and the definition of Rij , we verify that Sij {aij } =a ¯ij . 2,2 24 24 ˜2,x ˜2 22 2 20 2 Furthermore, it holds Sij,kl{aijkl} =¯aijkl. Hence, denoting L = L + ∂i bij ∂j ∂t · − b ∂t , we have ˜2,x 2 24 2 2 22 21 2 22 L v = ∂ij a¯ijkl∂klv − ∂i Sij {aij }∂j v + ai + ∂m(Smi{ami}) ∂iv. (3.25) 21 2 22 22 We claim that ai + ∂m(Smi{ami}) = 0. Indeed, using (3.24d), the form of Rij , and the symmetry of p and a0, we compute 2 22 2 23 23 23 22 10 2 12 10 2 0 0 0 0 0 20 0 Sij {aij } = Sij ∂n(pjin − pnij − pinj ) + pij + b Sij {aij }− (b ) aij + δ∂napi∂panj − δ∂p apn∂naij + b aij . 11 2 12 Note that we have seen in the proof of Proposition 3.3 that ai + ∂m(Smi{ami}) = 0. Using then (3.24f), direct computations lead to 21 2 22 2 0 0 0 0 2 0 0 ai + ∂m(Smi{ami})= δ∂m ∂npapiamn + δ∂m ∂napm∂pani − δ∂mp apn∂nami =0, 12 A. ABDULLE AND T. POUCHON which proves the claim. Combined with (3.25), the claim concludes the proof of the lemma. 3.3. Including a non-zero right hand side. To reach the accuracy Rε = O(ε3) in (3.21), we still have to remove the terms coming from the right hand side f. To do so, we let ϕ in (3.1) belongs to the unique class of solution ϕ of the equation

(∂2 + Aε)ϕ(t)= −Sεf(t) in W∗ (Ω) for a.e. t ∈ [0,T ε], t per (3.26) ϕ(0) = ∂tϕ(0) = [0],

ε · 0ε 0 · 1ε 1 · where, denoting χi = χi ·, ε ,θij = θij ·, ε ,θi = θi ·, ε , Sε ε 10 2 0ε 2 1ε ε 10 10 2 22 20 f = [ε(χi ∂if − b f)+ ε (θij ∂ij f + θi ∂if − χi ∂i(b f) + (b ) f + ∂i(bij ∂j f) − b f)].

∞ ε ∞ ε 2 2 2 ε ∗ We verify that ϕ ∈ L (0,T ; Wper(Ω)), ∂tϕ ∈ L (0,T ; L (Ω)) and ∂t ϕ ∈ L (0,T ; Wper(Ω)). Furthermore, the standard energy estimate for the wave equation ensures that

kϕkL∞(0,T ε;W) ≤ k∇xϕkL∞(0,T ε;L2(Ω)) ≤ CεkfkL1(0,T ε;H2(Ω)), (3.27) where C depends only on

22 10 20 0 1 λ, Λ, kχikC0(Ω;¯ C0(Y )), kbij kC1(Ω)¯ , |b |, |b |, kθij kC0(Ω;¯ C0(Y )), kθi kC0(Ω;¯ C0(Y )).

3.4. Proof of Theorem 3.1. To conclude this section, let us prove Theorem 3.1. The adaptation Bεu˜ is defined by (3.1), where u1,...,u4 are defined in (3.4), (3.7), (3.10), and (3.18), and ϕ ∈ ϕ solves (3.26). Then, combining Lemma 3.2 with Proposition 3.3, we verify that L1u˜ = L˜1u˜, where the tensors involved in the definition of L˜1u˜ satisfy the constraints (3.15). Hence, the cell problems (3.14) are well-posed and u3 is well defined. Similarly, combining Lemma 3.4 with Proposition 3.5, we verify that L2u˜ = L˜2u˜ and the definition of L˜2 ensures that u4 is well defined. Note that thanks to assumption (1.11), we verify that x 7→ Bεu˜(t, x) is Ω-periodic. This proves the existence of the adaptation Bεu˜. By construction (see (3.3)), Bεu˜ satisfies the properties (3.2) and the proof of the theorem is complete. 4. Proof of the main result. In this section, we prove the main result of the paper, Theorem 2.3. The proof is structured as follows. First, we use the correctors introduced in Section 3 to define the adaptation operator Bε. This operator satisfies Bεu˜(t) = [Bεu˜(t)], where Bεu˜ is defined in (3.1) and [·] denotes the 1 R equivalence class in Wper(Ω) = Hper(Ω)/ . We emphasize that in the proof, we work in the quotient Wper(Ω) because Bεu˜(t) does not have zero mean (alternatively, we can normalize all the non zero mean terms in ε ε B u˜(t) and work in Wper(Ω)). Next, using that u − u˜ ∈ Wper(Ω), we split the error as

ε ε ε ε ε ku − u˜kL∞(W ) = k[u − u˜]kL∞(W) ≤kB u˜ − [u ]kL∞(W) + k[u˜] − B u˜kL∞(W), (4.1) and estimate both terms. In particular, we prove that Bεu˜ satisﬁes the same equation as [uε] up to a remainder of order O(ε3) (Lemma 4.1). Consider the correctors

0 1 0 1 2 0 1 2 3 χi(x),θij (x),θi (x),κijk (x),κij (x),κi (x),ρijkl(x),ρijk (x),ρij (x),ρi (x) ∈ Wper(Y ), deﬁned as the solutions of the cell problems (3.5), (3.8), (3.14) and (3.22), and let ϕ be the solution of (3.26). Propositions 3.3 and 3.5 ensure that L1u˜ = L˜1u˜ and L2u˜ = L˜2u˜, where the deﬁnitions of the tensors in L˜1 (resp. L˜2) guarantee the well-posedness of the cell problems (3.14) (resp. (3.22)). Let us investigate the regularity of the correctors. Using regularity results for elliptic PDEs (see e.g. [18]), we can prove the following implications, for n ≥ 0, m ≥ 0 (see [41, Chapter 6]):

0 0 0 n ¯ m+1 n ¯ m,∞ χi,θij ,κijk,ρijkl ∈C (Ω; H (Y )) ⇐ a ∈C (Ω; W (Y )), 1 1 1 n ¯ m+1 n ¯ m,∞ n+1 ¯ {m−1}+,∞ θi ,κij ,ρijk ∈C (Ω; H (Y )) ⇐ a ∈C (Ω; W (Y )) ∩C (Ω; W (Y )), 2 2 n ¯ m+1 2 n+k ¯ {m−k}+,∞ κi ,ρij ∈C (Ω; H (Y )) ⇐ a ∈ ∩k=0C (Ω; W (Y )), 3 n ¯ m+1 3 n+k ¯ {m−k}+,∞ ρi ∈C (Ω; H (Y )) ⇐ a ∈ ∩k=0C (Ω; W (Y )), (4.2) EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 13

0 24 22 n ¯ n ¯ ∞ aij , a¯ijkl,bij ∈C (Ω) ⇐ a ∈C (Ω;L (Y )), 12 n ¯ n+1 ¯ ∞ a¯ij ∈C (Ω) ⇐ a ∈C (Ω;L (Y )), 22 n ¯ n+2 ¯ ∞ a¯ij ∈C (Ω) ⇐ a ∈C (Ω;L (Y )), where {·}+ = max{0, ·}. In particular, under the assumption of Theorem 2.3, all the correctors belong to 1 ¯ 3 2 ¯ 2 2 0 ¯ C (Ω; H (Y )) ∩C (Ω; H (Y )). As d ≤ 3, the Sobolev embedding Hper(Y ) ֒→Cper(Y ) holds and the correctors 1 ¯ 1 ¯ 2 ¯ 0 ¯ belongs to C (Ω; Cper(Y ))∩C (Ω; Cper(Y )). Hence, the following estimates hold (needed in the proof of Lemma 4.1 below)

0 1 0 1 max kχikC0(C0), kθij kC1(C1), kθi kC0(C0), kκijkkC2(C0), kκij kC2(C0), ijkl n 2 0 1 2 3 kκi kC2(C0), kρijklkC1(C1), kρijkkC1(C1), kρij kC1(C1), kρi kC1(C1), 12 10 24 22 22 20 ka¯ij kC2 , |b |, ka¯ijklkC3 , kbij kC2 , ka¯ij kC2 , |b | o ≤ C1(a, λ, Y )+ δC2(a, λ, Y ), (4.3) where Ci(a, λ, Y ) depend only on λ, Y , kakC1(W2,∞), kakC2(W1,∞), and kakC4(L∞), and δ is the parameter. Let us introduce the following useful application of the Green formula (see [8] for a proof): for c ∈ 1,∞ d 1 [Wper (Ω)] , v ∈ Hper(Ω), and w ∈Wper(Ω), we have

w w w [cm∂mv], L2 = − [∂mcmv], L2 − cm, ∂m L2 , (4.4) d where we recall that ∂mcm = m=1 ∂mcm. In order to shorten the notation, we deﬁne the following functions 1 ¯ ε · 0ε 0 · 1ε 1 · 0ε 1ε 2ε 0ε 1ε 2ε 3ε of Cper(Ω): χi = χi ·, ε , θijP= θij ·, ε , θi = θi ·, ε , and similarly κijk,κij ,κi ,ρijkl,ρijk,ρij ,ρi . We Bε 3 ∗ 3 deﬁne then the operators i : Hper(Ω) →W per(Ω) for v∈ Hper(Ω) as

Bε w w 0v, = [v], L2 , Bε w ε w 1v, = ε[χi ∂iv], L2 , Bε w 2 0ε 1ε w 2 0ε w 2v, = ε [(−∂mθmi + θi )∂iv], L2 − ε θmi∂iv, ∂m L2 , Bε w 3 0ε 3 1ε 2 2ε w 3v, = ε [κijk∂ijkv + κij ∂ij v + κi ∂iv], L2 , Bε w 4 0ε 1ε 3 2ε 2 3ε w 4 0ε 3 w 4v, = ε [(−∂mρmijk + ρijk)∂ijk v + ρij ∂ij v + ρi ∂iv], L2 − ε ρmijk∂ijkv, ∂m L2 , ∗ where h·, ·i denotes the dual evaluation h·, ·iWper ,Wper . The adaptation operator is then deﬁned as

4 Bε 2 ε 3 2 ε ∗ Bε Bε :L (0,T ; Hper(Ω)) → L (0,T ; Wper(Ω)), v 7→ v(t)= i (v(t)) + ϕ(t). (4.5) iP=0 2 ε 1 5 Bε Bε Note that if v ∈ L (0,T ; Hper(Ω) ∩ H (Ω)), then v(t) ∈Wper(Ω) and, using (4.4), we verify that u˜(t)= ε ε k 4 [B u˜(t)], where B u˜ is defined in (3.1) (with {u }k=1 defined in (3.4), (3.7), (3.10), and (3.18)). For ε ε A = −∇x · a (x)∇x · , we thus define ε ε ε ε A B u˜(t), w ∗ = A [B v(t)], w ∗ . Wper,Wper Wper,Wper

Remark that the definition of Bε in (4.5) allows to consider functions with lower regularity than Bε. In Bε 2 2 ∞ ε 3 particular, (∂t u˜) is well-defined, as ∂t u˜ ∈ L (0,T ; Hper(Ω)). This is needed to prove the following lemma. Lemma 4.1. Under the assumptions of Theorem 2.3, Bεu˜ satisfies

2 ε Bε Rε ∗ ε (∂t + A ) u˜(t)= [f(t)] + u˜(t) in Wper(Ω) for a.e. t ∈ [0,T ], Rε ∞ ε ∗ where the remainder u˜ ∈ L (0,T ; Wper(Ω)) is given as Rε Rε Rε u˜(t), w ∗ ∗ = ( u˜)0(t), w 2 + ( u˜)1(t), ∇xw 2 , Wper,Wper L L 14 A. ABDULLE AND T. POUCHON with the bound ε ε k(R u˜)0kL∞(0,T ε;L2(Ω))+k(R u˜)1kL∞(0,T ε;L2(Ω)) 3 5 ∞ 2 ∞ ≤ Cε k=1 |u˜|L (0,T ε;Hk(Ω)) + k∂t u˜kL (0,T ε;H3(Ω)) , P for a constant C that depends only on λ, Y , kakC1(Ω;W¯ 2,∞(Y )), kakC2(Ω;W¯ 1,∞(Y )), kakC4(Ω;L¯ ∞(Y )), and δ. ε Rε Proof. Let us denote h·, ·iWper ,Wper as h·, ·i. For a fixed t ∈ [0,T ], we compute the remainder u˜(t)= 2 ε Bε 2Bε (∂t + A ) u˜(t) − [f(t)]. Let us first compute explicitly the first term, ∂t u˜(t). For the sake of clarity, we Bε 2Bε 2 Bε 2 drop the notation of the evaluation in t. From the definition of in (4.5), it holds ∂t u˜ = i=0 i ∂t u˜ + 2 Rε Rε 4 Bε 2 ∂t ϕ + 1u˜, where 1u˜ = i=3 i ∂t u˜, i.e., P P 2Bε w 2 w ε 2 2 0ε 1ε 2 w ∂t u,˜ = [∂t u˜], L2 + [εχi ∂i∂t u˜ + ε (−∂mθmi + θi )∂i∂t u˜], L2 2 0ε 2 2 Rε − ε θmi∂i∂t u,˜ ∂mw + ∂t ϕ + 1u,˜ w . (4.6) We rewrite the three first terms of the right hand side. Note that thanks to the regularity ofu ˜ and the effective equation (2.9), we have the following equalities 2 0 ˜1 2 ˜2 2 ∂t u˜ = f + ∂m(amn∂nu˜) − εL u˜ − ε L u˜ in L0(Ω), (4.7) 2 2 0 ˜1 2 ˜2 2 ∂i∂t u˜ = ∂if + ∂im(amn∂nu˜) − ε∂i(L u˜) − ε ∂i(L u˜) in L (Ω). (4.8) Using (4.7), we rewrite the first term of (4.6) as 2 0 ˜1,x 10 0 2 ˜2 10 ˜1 [∂t u˜] = [f] + [∂m(amn∂nu˜)+ ε − L u˜ − b ∂m(amn∂nu˜) + ε − L u˜ + b L u˜ ] + [−εb10f + ε3b10L˜2u˜],

˜1,x ˜1 10 2 ˜1 ˜2 where L = L − b ∂t . Using the deﬁnitions of L and L and (4.7), we have 2 ˜2 10 ˜1 w ε [−L u˜ + b L u˜], L2 2 ˜2,x 10˜1,x 10 2 20 0 w = ε [−L u˜ − b L u˜ + ((b ) − b )∂m(amn∂nu˜)], L2 2 22 2 w 2 10 2 20 ˜1 2 ˜2 w − ε bmi∂i∂t u,˜ ∂m L2 + ε [((b ) − b )(f + εL u˜ + ε L u˜)], L2 , ˜2,x ˜2 22 2 20 2 where L = L + ∂i(bij ∂j ∂t ·) − b ∂t . We thus obtain

2 w 0 ˜1,x 10 0 [∂t u˜], L2 = [f] + [∂m(amn∂nu˜)+ ε − L u˜ − b ∂m(amn∂nu˜) 2 ˜2,x 10 ˜1,x 10 2 20 0 w + ε − L u˜ + b L u˜ + ((b ) − b )∂m(amn∂nu˜)], L2 2 22 2 w Sε Rε w − ε bmi∂i∂t u,˜ ∂m L2 + 1f + 2u,˜ L2 , (4.9) where Sε 10 2 10 2 20 1f = [−εb f + ε ((b ) − b )f], Rε 3 10 ˜2 3 10 2 20 ˜1 ˜2 2u˜ = [ε b L u˜ + ε ((b ) − b )(L u˜ + εL u˜)]. Next, we use (4.8) and then (4.7) to write the second term of (4.6) as ε 2 ε 2 0 2 ε ˜1,x 2 ε 10 0 Sε Rε [εχi ∂i∂t u˜] = [εχi ∂im(amn∂nu˜) − ε χi ∂i(L u˜) − ε χi ∂i(b ∂m(amn∂nu˜))] + 2f + 3u,˜ (4.10) where Sε ε 2 ε 10 2f = [εχi ∂if − ε χi ∂i(b f)], Rε 3 ε ˜2 3 ε 10 ˜1 ˜2 3u˜ = [−ε χi ∂i(L u˜)+ ε χi ∂i(b (L u˜ + εL u˜))]. Furthermore, using (4.8) and formula (4.4), we rewrite 2 0ε 2 w 2 0ε 22 2 w − [ε ∂mθmi∂i∂t u˜], L2 − ε (θmi + bmi)∂i∂t u,˜ ∂m L2 2 0ε 3 0 22 2 0 w Sε Rε w = [ε θij ∂ijm(amn∂nu˜)+ ∂i(bij ∂jm(amn∂nu˜))], L2 + 3f + 4u,˜ . (4.11) EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 15 where

Sε w 2 0ε 2 22 w 3f, = ε [θij ∂ij f + ∂i(bij ∂j f)], L2 , Rε w 3 0ε ˜1 ˜2 w 3 0ε 22 ˜1 ˜2 w 4u,˜ = ε [∂mθmi∂i(L u˜ + εL u˜)], L2 + ε (θmi + bmi)∂i(L u˜ + εL u˜), ∂m L2 , and, using (4.8), we rewrite 2 1ε 2 2 1ε 2 0 Sε Rε [ε θi ∂i∂t u˜] = [ε θi ∂im(amn∂nu˜)] + 4f + 5u,˜ (4.12)

Sε 2 1ε Rε 3 1ε ˜1 ˜2 where 4f = [ε θi ∂if] and 5u˜ = [ε θi ∂i(L u˜ + εL u˜)]. Combining equalities (4.6), (4.9), (4.10), (4.11) and (4.12), we ﬁnally obtain 2Bε 0 2 0 ˜1,x 10 0 ∂t u˜ = [f] + [∂m(amn∂nu˜)] + ε[χi∂im(amn∂nu˜) − L u˜ − b ∂m(amn∂nu˜)] 2 0ε 3 0 1ε 2 0 ε ˜1,x ε 10 0 + ε [θij ∂ijm(amn∂nu˜)+ θi ∂im(amn∂nu˜) − χi ∂i(L u˜) − χi ∂i(b ∂m(amn∂nu˜)) ˜2,x 10 ˜1,x 10 2 20 0 22 2 0 − L u˜ + b L u˜ + ((b ) − b )∂m(amn∂nu˜)+ ∂i(bij ∂jm(amn∂nu˜))] 2 4 Sε 5 Rε + ∂t ϕ + i=1 i f + i=1 i u.˜ (4.13) P P εBε x For the second term, A u˜(t), we have (the correctors and a are evaluated at x, y = ε ) AεBεu˜ = −1 [ ε − ∇y · (a(∇yχi + ei)) ∂iu˜ 0 0 T 2 + ε − ∇y · (a(∇yθij + eiχj )) − ei a(∇yχj + ej) ∂ij u˜ 0 1 + ε − ∇y · (a(∇yθi + ∇xχi)) − ∇x · (a(∇yχi + ei)) ∂iu˜ 1 0 0 T 0 3 + ε − ∇y · (a(∇yκijk + eiθjk)) − ei a(∇yθjk + ejχk) ∂ijk u˜ 1 1 0 1 0 T 1 2 + ε − ∇y · (a(∇yκij + ∇xθij + eiθj )) − ∇x · (a(∇yθij + eiχj )) − ei a(∇yθj + ∇xχj ) ∂ij u˜ 1 2 1 1 + ε − ∇y · (a(∇yκi + ∇xθi )) − ∇x · (a(∇yθi + ∇xχi)) ∂iu˜ 2 0 0 T 0 0 4 + ε − ∇y · (a(∇yρijkl + eiκjkl)) − ei a(∇yκjkl + ej θkl) ∂ijklu˜ 2 1 0 1 0 0 + ε − ∇y · (a(∇yρijk + ∇xκijk + eiκjk)) − ∇x · (a(∇yκijk + eiθjk)) T 1 0 3 − ei a(∇yκjk + ∇xθjk) ∂ijku˜ 2 2 1 2 1 0 1 + ε − ∇y · (a(∇yρij + ∇xκij + eiκj )) − ∇x · (a(∇yκij + ∇xθij + eiθj )) T 2 1 2 − ei a(∇yκj + ∇xθj ) ∂ij u˜ 2 3 2 2 1 + ε − ∇y · (a(∇yρi + ∇xκi )) − ∇x(a(∇yκi + ∇xθi )) ∂iu˜ ] ε Rε Rε + A ϕ + 6u˜ + 7u,˜ (4.14) where, deﬁning the following functions of (x, y),

0 0 0 1 1 0 1 Rijkl = a(∇yρijkl + eiκjkl), Rijk = a(∇yρijk + ∇xκijk + eiκjk), 2 2 1 2 3 3 2 Rij = a(∇yρij + ∇xκij + eiκj ), Ri = a(∇yρi + ∇xκi ), Rε Rε the remainders 6u˜ and 7u˜ are given by Rε 3 T 0 5 0 4 T 1 4 1 3 6u˜ = ε [emRijkl∂mijklu˜ + ∇x · Rijkl∂ijklu˜ + emRijk∂mijk u˜ + ∇x · Rijk∂ijk u˜ T 2 3 2 2 T 3 2 3 + emRij ∂mij u˜ + ∇x · Rij ∂ij u˜ + emRi ∂miu˜ + ∇x · Ri ∂iu˜], Rε w 4 T 0 4 1 3 2 2 3 w 7u,˜ = ε ema(∇xρijkl∂ijklu˜ + ∇xρijk∂ijku˜ + ∇xρij ∂ij u˜ + ∇xρi ∂iu,˜ ∂m L2 4 0 5 1 4 2 3 3 2 w + ε amnρijkl∂nijklu˜ + amnρijk∂nijk u˜ + amnρij ∂nij u˜ + amnρi ∂niu,˜ ∂m L2 . Combining now (4.13) and (4.14), and using cell problems (3.5), (3.8), (3.14), (3.22), and the deﬁnition of ϕ 4 Sε Sε Rε 7 Rε in (3.26) (verify that i=1 i f = f ), we obtain the remainder u˜ = i=1 i u˜. Using (4.3),we verify ε that R u˜ satisﬁes estimateP (4.6) and the proof of the lemma is complete. P 16 A. ABDULLE AND T. POUCHON

Let us recall the following error estimate, proved in [8]. ∞ ε ∞ ε 2 2 Lemma 4.2. Assume that η ∈ L (0,T ; Wper(Ω)), with ∂tη ∈ L (0,T ; L (Ω)), ∂t η ∈ 2 ε ∗ L (0,T ; Wper(Ω)) satisﬁes

2 ε ∗ ε ∂t η(t)+ A η(t)= r(t) in Wper(Ω) for a.e. t ∈ [0,T ], 0 1 (4.15) η(0) = η , ∂tη(0) = η ,

0 1 2 2 ε ∗ where η ∈Wper(Ω), η ∈ L (Ω), and r ∈ L (0,T ; Wper(Ω)) is given as

r(t), w ∗ = r0(t), w 2 + r1(t), ∇xw 2 , Wper(Ω),Wper(Ω) L (Ω) L (Ω) 2 ε 2 2 ε 2 d with r0 ∈ L (0,T ; L (Ω)) and r1 ∈ [L (0,T ;L (Ω))] . Then the following estimate holds

1 0 −2 kηkL∞(0,T ε;W) ≤ C(λ) kη kL2(Ω) + kη kL2(Ω) + ε T kr0kL∞(0,T ε;L2(Ω)) + kr1kL∞(0,T ε;L2(Ω)) , (4.16) where C(λ) depends only on the ellipticity constant λ and the norm k·kW is defined in (1.7). We now have all the tool to prove the theorem. Proof of Theorem 2.3. Let us estimate the two terms of the right hand side in (4.1). First, note that η = [uε] − Bεu˜ satisfies 2 ε Rε ∗ ε Rε (∂t + A )η(t) = u˜(t) in Wper(Ω) for a.e t ∈ [0,T ], where u˜ is defined in Lemma 4.1. Hence, using Lemma 4.2, the first term in (4.1) satisfies

ε ε 1 0 5 B ∞ ∞ 2 ∞ k[u ] − u˜kL (W) ≤ Cε kg kH4 + kg kH4 + k=1 |u˜|L (Hk ) + k∂t u˜kL (H3) , (4.17) P where C depends on T , λ, Y , kakC1(Ω;W¯ 2,∞(Y )), kakC2(Ω;W¯ 1,∞(Y )), kakC4(Ω;L¯ ∞(Y )), and δ. Next, using the deﬁnition of Bε (4.5) and the estimates (3.27) and (4.3), the second term of (4.1) satisﬁes

ε 5 B ∞ ∞ k u˜ − [u˜]kL (W) ≤ Cε k=1 |u˜|L (Hk) + kfkL1(H2) , (4.18) P where C depends on λ, Y , kakC1(Ω;W¯ 2,∞(Y )), kakC2(Ω;W¯ 1,∞(Y )), kakC4(Ω;L¯ ∞(Y )), and δ. Combining (4.1), (4.17), and (4.18), we obtain (2.10) and the proof of the theorem is complete. 5. A priori error analysis of the FE-HMM-L in locally periodic media. The FE-HMM-L is a numerical homogenization method for the long time approximation of the wave equation introduced in [5, 4]. ε In [9], a priori estimates for the long time error between u and the approximation uH were proved in one dimension for uniformly periodic tensors. In this section, thanks to Theorem 2.3, we provide a complement to this analysis as we present error estimates that hold in the locally periodic case (again in one dimension). This result is valid in small domains. In addition, we prove a new a priori error estimate that holds in arbitrarily large domain. Let us first express the family of effective equation in the specific one-dimensional case. Let x ∈ Ω be fixed and recall that hχ(x)iY = 0. As a(x, ·)(1 + ∂yχ(x, ·)) ∈ H(div, Y ), using integration by parts and equation (2.1a), we obtain for any y1,y2 ∈ Y ,

y2 a(x, y)(∂yχ(x, y)+1) = − Hy (y) − Hy (y) ∂y a(x, y)(∂yχ(x, y)+1) dy =0, Y 2 1 y=y1 R where Hy is the Heaviside step function centered in y. Hence, the function y 7→ a(x, y)(∂yχ(x, y) + 1) is constant. The deﬁnition of a0 in (2.2) then implies that

0 a(x, y)(∂yχ(x, y)+1)= a (x) ∀(x, y) ∈ Ω × Y. (5.1)

0 A similar argument, using (2.1b), (2.1c), and the fact that a (x)=1/ 1/a(x, ·) Y , leads to

0 1 a(x, y) ∂yθ (x, y)+ χ(x, y) =0, a(x, y) ∂yθ (x, y)+ ∂xχ(x, y) =0 ∀(x, y) ∈ Ω × Y. (5.2) EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 17

Thanks to (5.1), we verify that the coefficients defined in (2.4) satisfy p13(x)= q12(x) = 0 and thusa ¯12(x)= b10 = 0. Similarly, using (5.2) in (2.6) and (2.8), we verify thata ˇ24(x)= p23(x)= p22(x) = 0. Hence, in one dimension, the family E (Definition 2.1) is constituted of the equations

2 0 2 2 24 2 22 2 22 20 2 ∂t u˜ − ∂x(a ∂xu˜)+ ε ∂x(a ∂xu˜) − ∂x(b ∂x∂t u˜) − ∂x(a ∂xu˜)+ b ∂t u˜ = f, where the coeﬃcients are deﬁned for some parameter r ≥ 0 as

a24(x)= ra0(x)2, b22(x)= χ(x)2 + ra0(x), Y (5.3) 20 2 0 22 0 2 0 20 0 b = r maxx∈Ω{∂xa (x)}, a (x)= −ra (x)∂xa (x)+ b a (x).

2 22 2 In particular, the equation corresponding to the choice r = 0 involves the single correction −ε ∂x(b ∂x∂t ·). This is precisely the effective model on which the FE-HMM-L relies (see [5, 4, 9]). Let us briefly recall the definition of the FE-HMM-L. Let TH be a partition of Ω of size H. For ℓ ∈ N>0, the macro finite element space is defined as

ℓ VH (Ω) = {vH ∈ Wper(Ω) : vH |K ∈P (K) ∀K ∈ TH }, (5.4)

′ ℓ J ′ ′ J where P (K) is the space of polynomials on K of degree at most ℓ. Let {ωˆj, xˆj }j=1 and {ωˆj, xˆj }j=1 be the quadrature formulas used for the construction of the stiffness and mass matrices, respectively. We assume that these formulas satisfy the requirements that ensure the optimal convergence rates of the FEM with numerical quadrature (see [21, 20] or [1]). For every macro element K ∈ TH and every j ∈ {1,...,J}, we define the sampling domain Kδj = xKj + δY , where δ ≥ ε. Each sampling domain Kδj is discretized into a partition Th of size h. For q ∈ N>0, the micro finite element space is defined as

q Vh(Kδj )= {zh ∈ Wper(Kδj ): zh|Q ∈P (Q) ∀Q ∈ Th}. (5.5)

We deﬁne the following bilinear forms: for vH , wH ∈ VH (Ω),

J ωK j x AH (vH , wH )= a xKj , ε ∂xvh,Kj (x)∂xwh,Kj (x) dx, (5.6) |Kδj | ZKδ KX∈TH Xj=1 j

vH , wH Q = vH , wH H + vH , wH M , (5.7) ′ J v , w = ω′ v (x′ )w (x′ ), (5.8) H H H Kj H Kj H Kj KX∈TH Xj=1 J ωK v , w = j v − vlin w − wlin (x) dx, (5.9) H H M h,Kj H,Kj h,Kj H,Kj |Kδj | ZKδ KX∈TH Xj=1 j where the piecewise linear approximation of vH (resp. wH ) around xKj is given by

vlin (x)= v (x ) + (x − x )∂ v (x ), H,Kj H Kj Kj x H Kj and the micro functions vh,Kj for vH (resp. wH ) are the solutions of the following micro problems in Kδj : ﬁnd v such that (v − vlin ) ∈ V (K ) and h,Kj h,Kj H,Kj h δj

· a xK , ∂xvh,K , ∂xzh 2 =0 ∀zh ∈ Vh(Kδj ). (5.10) j ε j L (Kδj ) ε x We emphasize that in (5.6) and (5.10), the tensor is collocated in the slow variable, i.e., a (x) = a xKj , ε ε ∀x ∈ Kδj . The approximation of the FE-HMM-L is uH : [0,T ] → VH (Ω) such that

2 ε ∂t uH (t), vH Q + AH uH (t), vH = f(t), vH L2 ∀vH ∈ VH (Ω) for a.e. t ∈ [0,T ], 0 1 (5.11) uH (0) = IH g, ∂tuH(0) = IH g, 18 A. ABDULLE AND T. POUCHON where IH is an interpolation operator onto VH (Ω) satisfying the optimal convergence rates (e.g. the nodal interpolation operator deﬁned in [20]). Let us now combine Theorem 2.3 with the a priori error analysis from [9]. Assume that δ/ε ∈ N>0, q = 1, and that the data are suﬃciently regular. Then the following a priori error estimate holds:

2 ℓ+1 ℓ 4 ε h H H k ku − u k ∞ ε 2 ≤ C ε + + + k∂ u¯k ∞ ℓ+1 , (5.12) H L (0,T ;L (Ω)) ε2 ε2 ε t L (H ) kP=0 where C is independent of H, h, ε, and δ but depends on Ω. Note that the dependence of the constant on the domain Ω is an issue when the method is used in pseudoinfinite domains. Indeed, as we consider timescales O(ε−2), a pseudoinfinite domain must have a diameter of order O(ε−2) (if the homogenized wave speed is of order O(1)) and (5.12) can not be used. This issue is settled by the following new result. Theorem 5.1. Let u¯ denote the effective solution in the family that corresponds to the parameter ℓ ∞ 0 1,∞ r = 0 in (5.3). Assume that δ/ε ∈ N>0, h ≤ ε, and q = 1. If a ∈ C (Ω;L¯ (Y )) ∩C (Ω;¯ W (Y )) and k ∞ ε ℓ+1 ∂t u¯ ∈ L (0,T ; H (Ω)) for 0 ≤ k ≤ 4, then the error e =u ¯ − uH satisfies the estimate

2 ℓ ℓ+1 4 data h H k k∂ ek ∞ 2 + k∇ ek ∞ 2 ≤ C e 1 + + |u¯| ∞ σ + k∂ u¯k ∞ ℓ+1 , (5.13) t L (L ) x L (L ) H ε2 ε2 L (H ) t L (H ) σP=1 kP=1

data 0 0 1 1 ˜ ∞ ∞ ˜ where eH1 = |g − IH g |H1(Ω) + kg − IH g kH1(Ω) and C = C kakCℓ(L ) + kakC0(W1, ) with C independent of ε, H and Ω. We emphasize that the constant C˜ is independent of Ω. Hence, combining Theorems 2.3 and 5.1, we obtain (if the data are suﬃciently regular and have a spatial support of order O(1))

2 ℓ ε h H ku − u k ∞ ε = O ε + + , (5.14) H L (0,T ;W ) ε2 ε2 where the norm k·kW is deﬁned in (1.8). In particular, estimate (5.14) can be used in pseudoinﬁnite domains. To discuss the computational cost of the FE-HMM-L, let us compare it with a standard P1-FEM ap- ε proximation of u , denoted uh. As argued in [9] (see also [7]), the optimal error estimate for uh is given by (if the initial data are well-prepared)

ε h ku − u k ∞ ε 2 ≤ C . (5.15) h L (0,T ;L (Ω)) ε3 Note that in (5.15), the constant C depends on the domain (through the Poincaréconstant and the constant in the elliptic regularity estimate). Hence, in a large domain of diameter O(ε−2), (5.15) scales worse with respect to ε. On the contrary, we emphasize that (5.14) holds independently of the domain. For the sake of comparison, let us then consider a small domain of diameter O(1). We fix an order of tolerance τ for the error and, based on the corresponding error estimate, we compute the cost of each method. Let us denote cost(∆t,N) the cost per time-step of the time integration of a second order ODE of dimension N. From (5.15), it follows that the cost of the P1-FEM is cost(∆t,ε−3τ −1). The cost of the FE-HMM-L, based on (5.14), is cost(∆t,ε−2/ℓτ −1/ℓ), to which we add the offline cost of the resolution of the micro problems, H−1(ε/h) = ε−1−2/ℓτ −1/2−1/ℓ. Hence, the FE-HMM-L offers a significant reduction of the computational cost. Furthermore, note that in the FE-HMM-L, the macro FE degree ℓ can be increased to reduce the cost. This is not the case for the fine scale FEM as higher order FE do not improve (5.15) (negative powers of ε appear from the higher derivatives of uε). Let us prove Theorem 5.1 (a detailed proof is provided in [41, Chapter 7]). We follow the technique of elliptic projection as done in [27, 14, 15]. We split the error as

u¯ − uH = (¯u − πH u¯) − (uH − πH u¯)= η − ζH , (5.16)

ε where πH u¯ is a new elliptic projection. The following deﬁnition of πH u¯ : [0,T ] → VH (Ω) is the key to avoid ε the dependence of the constant on Ω: for a.e. t ∈ [0,T ], πH u¯(t) ∈ VH (Ω) satisﬁes

2 πH u¯(t), vH Q + AH πH u¯(t), vH = f(t), vH L2 − IH ∂t u¯(t), vH Q + IH u¯(t), vH Q, (5.17) EFFECTIVE MODELS FOR LONG TIME WAVE PROPAGATION IN LOCALLY PERIODIC MEDIA 19 for all test functions vH ∈ VH (Ω). Using the same technique as in [9], we prove the following lemma. k k+2 ∞ ε ℓ+1 k ∞ ε 1 Lemma 5.2. Assume that ∂t u,¯ ∂t u¯ ∈ L (0,T ; H (Ω)) for k ≥ 0. Then ∂t πH u¯ ∈ L (0,T ; H (Ω)) and η =u ¯ − πH u¯ satisﬁes

k k kIH ∂t ηkL∞(H1) + k∂t ηkL∞(H1) ℓ+1 k+2 (5.18) 2 ℓ k ∞ 2 ℓ ∞ ≤ C (h/ε) + H σ=1 |∂t u¯|L (Hσ) + ε(h/ε) + H k∂t u¯kL (Hℓ+1) , P ˜ ∞ ˜ where C = C kakCℓ(L∞) + kakC0(W1, ) with C independent of H, ε, and Ω. In [9, Lemma 3.11], a similar estimate is proved, with the major difference that the constant involved in the estimate depends on the Poincaréconstant. In the proof of Lemma 5.2, the need of the Poincaré inequality is avoided thanks to the definition of the new elliptic projection πH u¯ in (5.17). The following lemma is proved in [9]. Lemma 5.3. The following estimate holds for ζH = uH − πH u¯,

∞ ∞ data ∞ ∞ k∂tζH kL (L2) + k∇xζH kL (H1) ≤ C eH1 + kηkL (H1) + k∂tηkL (H1) (5.19) −2 −2 2 + ε kIH ηkL∞(H1) + ε kIH ∂t ηkL∞(H1) , data 0 0 1 1 where eH1 = |g − IH g |H1 + kg − IH g kH1 and C is independent of H, ε and Ω. The splitting (5.16) combined with Lemmas 5.2 and 5.3 proves Theorem 5.1. 6. Numerical experiments. In this section, we illustrate numerically our main result. We consider a one-dimensional example and compare the heterogeneous solution with several eﬀective solutions of the family E, the homogenized solution, and the approximation of the FE-HMM-L. −20x2 Let us ﬁx the initial data and the right hand side for the test problem as g0(x)= e , g1 = 0, f = 0, and consider the locally periodic tensor given by

x 249 1 1 x a x, ε = 419 + 6 sin(2πx)+ 6 sin 2π ε , (6.1) 0 0 y with ε =1/20. We compute explicitly a (x)=1 1/a(x, ·) Y and χ(x, y)= a (x) 1/a(x, z) dz −y +C0(x), 0 R ε −2 where C0 ensures that hχ(x)iY = 0 ∀x ∈ Ω. We verify that Ω a (x) dx ≈ 3/4. We let T = ε = 400, ε 0 and compare u , u and effective solutionsu ˜r in the family, whereR p the subscript r specifies the dependence ofu ˜ on the parameter r in (5.3). For the waves not to reach the boundary, we consider the pseudoinfinite ε domain Ω = (−301, 301). To approximate u , we use a spectral method on a grid of size href = ε/25 and 0 the leap frog scheme for the time integration with time step ∆t = href /50. To approximate u andu ˜r, the same methods are used with h = ε/4 and ∆t = h/50. Note that the second order ODE obtained after the space discretization ofu ˜r is implicit. To solve it, we use a gradient method at each time iteration of the ε 0 leap frog scheme. In Figure 6.1, we display the frontal right going wave of u , u , andu ˜r for several values of r ∈ [0, 0.1] at t = ε−2 = 400. We observe that the macroscopic behavior of uε is not well described 0 ε by u . On the contrary,u ˜r describes well u for every value of the parameter r, as predicted by Theorem 2 ε 0 2.3. Let us now compare the L error between u (t) and u (t),u ˜r(t). We denote the normalized error as ε ε 0 err(v)(t)= k(u − v)(t)kL2(Ω)/ku (t)kL2(Ω). In Figure 6.2, we display err(u )(t) and err(˜ur)(t). We note that the error of the homogenized solution increases quickly with respect to t, confirming that at long times u0 ε does not describe well u . Next, we see that the error ofu ˜r increases notably as r increases. As Figure 6.1 showed, the frontal wave is well captured for all the values of r, hence the error for the large values of r is ε located elsewhere. In fact, we verify that away from the frontal waveu ˜r drives away from u as r increases. Hence, we conclude that the smaller r is the more accurateu ˜r is. Next, we compute the approximation provided by the FE-HMM-L, denoted by uH , for the model problem. We let the macro FE degree be ℓ = 3, the micro FE degree be q = 1, and δ = ε. Referring to (5.14), we 5/2 set H = ε and h = ε . In Figure 6.1, we observe that the approximation uH approximates accurately the ε ε macroscopic behavior of u . In particular, uH captures accurately the long time dispersive effects of u . Acknowledgements. This work was partially supported by the Fonds National Suisse, project No. 200021 150019.

REFERENCES 20 A. ABDULLE AND T. POUCHON

t = 400

0.2 r 0.097

0.078 0.1 0.059 0.041 0 uε 0.022 u0 0.003 u˜r −0.1 uH

294.5 295 295.5 296 296.5 297

ε 0 Fig. 6.1. Comparison between the frontal waves of u , u , u˜r and uH at time t = 400 and zoom on x ∈ [296.3, 296.9].

r 0.2 0.097 err(u0)(t) 0.078 err(u˜r)(t) 0.059 0.1 0.041 0.022 0.003 0 50 100 150 200 250 300 350 400 t

2 ε 0 Fig. 6.2. Comparison of the normalized L error between u and u , u˜r on the time interval [0, 400].

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INSTITUTE of MATHEMATICS MATHICSE Group

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CH-1015 Lausanne

2017

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