Keywords: Méthodes asymptotiques, équation des ondes, temporel, fentes minces, conditions de Kirchhoff

Étude de la propagation des ondes acoustiques dans une jonction de fentes minces

Résumé : Dans cet article, nous utilisons la théorie des développements asymptotiques raccordés pour analyser la propagation d'ondes acoustiques à travers une jonction de fentes minces. ceci nous permet de proposer des conditions de Kirchhoff améliorées pour le problème limite 1D. Ces conditions sont analysées et validées numériquement.

Introduction

One can consider time harmonic wave propagation and time domain propagation in thin domains that are junctions of thin slots whose thickness ε is small with respect to the wave length λ and converge, when ε tends to 0, to a 1dimensional graph. A contrario, one can imagine that we start from a 1D graph of straight lines that we thicken symmetrically so that we obtain a ε-thickness 2D graph (an example is given by the figure 1).

2D graph with thickness ε 1D graph On this 2D graph (that will be call Ω ε in the following), one wants to solve the following problem: find

u ε ∈ C 0 (R * + , H 1 (Ω ε )) ∩ C 1 (R * + , L 2 (Ω ε )) satisfying                        ∂ 2 u ε ∂t 2 (t, x) -∆u ε (t, x) = 0 for (t, x) ∈ R * + × Ω ε ∂u ε ∂ n (t, x) = 0 for (t, x) ∈ R * + × ∂Ω ε u ε (0, x) = f ε for x ∈ Ω ε ∂u ε ∂t (0, x) = g ε for x ∈ Ω ε (1)
with (f ε , g ε ) ∈ H 1 (Ω ε ) × L 2 (Ω ε ) suth that we assume, for ε small enough, that these functions vanish on the different junctions, and do not depend on the crosswise variable on the slots. The limit model is known for very long time but its justification seems to be very recent, with the works of Jacob Rubinstein, Michelle Schatzman [START_REF] Rubinstein | Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum[END_REF][START_REF] Rubinstein | Variational problems on multiply connected thin strips. II. Convergence of the Ginzburg-Landau functional[END_REF] and Peter Kuchment [START_REF] Kuchment | Graph models for waves in thin structures[END_REF]. In this limit model, the solution only depends of the topology of the graph and satisfy an one-dimensional time domain acoustic equation. Moreover, this solution is continuous at each node of the graph, and satisfy the Kirchhoff laws. To be more precise, the sum of the outgoing normal derivatives (with appropriate constants) is equal to zero. For studying the behaviour of u ε for small ε, we will use the method of matched asymptotics expansions. This is a well-known method that has been developed at the beginning of the 70's, initially to analyse boundary layer phenomena. This approach has been developed quasi-independently (one does not find a lot of cross citations in the publications) by two mathematical schools, from two rather different points of view (see also [START_REF] Il'in | Matching of asymptotic expansions of solutions of boundary value problems[END_REF][START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF] for more references). In this article, the use of the matched asymptotics expansions is based on a recent work of Patrick Joly and Sébastien Tordeux [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF]. As it is classical with this type of problem, the work can be naturally divided into three steps. The first one is related to obtaining the formal asymptotic expansion and constitutes the most algebraic and calculator part. The next two cover two different aspects of the mathematical justification and make use of various techniques for the analysis of PDE's.

• Step 1: Derive formal expansion. One starts from an ansatz, i.e. a form of a priori behaviour for the asymptotic expansion that is injected in the equations of the initial (ε dependant) problem, to deduce a series of (ε independent) elementary problem that are supposed to characterize the terms of the asymptotic expansion.

•

Step 2: Show that the various terms of the asymptotic expansions are well defined, i.e. that the above elementary problems are well posed. This is not necessary straightforward since these problems are quite often not standard.

• Step 3: Justify the asymptotic expansion, namely establish error estimates between the true solution and truncated asymptotic expansions.

A very particular case has been treated in [START_REF] Joly | Construction and analysis of improved kirchhoff conditions for acoustic wave propagation in a junction of thin slots[END_REF]: in this article, we consider the case of two slots of same width being connected by one junction. The outline of this paper is the following:

• in section 1, we explain the model problem we consider, and we claim the main results of this article,

• in section 2, we use the technic of matched asymptotics to give the development of the exact solution of the whole domain. One can remark that, for giving our approximated model, we only need the first terms of our development, but one can also see that the approximated model we need is the first order of a generalized family of approximated problems,

• in section 3, we justify the expansions of section 2, and we prove that we can build some approximate function which differs from the exact solution with an error as small as we want,

• finally in section 4, we explain how we build one 1D problem (in space) whose solution differs from the exact solution with an arror as small as we want.

INRIA 1 Model problem and main results

The model problem

In this section we introduce the geometry and the equations of our problem. We consider a domain made of the junction of N straight slots (see figure 2). More precisely, we consider the union of N thin rectangles and a junction zone. The i th rectangle has length L i and thickness c i ε, where ε is the small parameter in the analysis. A geometrical characteristic of this domain is the relative width of the rectangles, given by the numbers c i . For the analysis, we consider in fact a family of such thin domains denoted Ω ε with varying ε. We make the choice (this has an influence on the asymptotic analysis) that each rectangle of Ω ε expands symmetrically from a fixed segment S i whose one of the two vertices does not depend on i (see again figure 2).

J ε O t i n i
Ω ε = Ω ε 1 ∪ Ω ε 2 ∪ . . . ∪ Ω ε N ∪ J ε (2)
with J ε = εJ where J does not depend on ε, and

Ω ε i = x ∈ R 2 / 0 < x • t i < L i , -ciε 2 < x • n i < ciε 2
t i and n i are given by figure 2 (3)

The problem we consider is: find

u ε ∈ C 0 (R * + , H 1 (Ω ε ) ∩ C 1 (R * + , L 2 (Ω ε )
) such that (we recall here the wave system (1) we want to solve)

                       ∂ 2 u ε ∂t 2 (t, x) -∆u ε (t, x) = 0 for (t, x) ∈ R * + × Ω ε ∂u ε ∂ n (t, x) = 0 for (t, x) ∈ R * + × ∂Ω ε u ε (0, x) = f ε for x ∈ Ω ε ∂u ε ∂t (0, x) = g ε for x ∈ Ω ε
For this system, we can define the associate energy

E ε (t, u ε ) = 1 2ε Ω ε |∇u ε (t, x)| 2 + ∂u ε ∂t (t, x) 2 dx (4)
and we have (thanks to the Neumann boundary condition)

E ε (t, u ε ) = E ε (0, u ε ) = 1 2ε Ω ε |∇f ε (x)| 2 + |g ε (x)| 2 dx (5)
In this energy, we add the ε -1 term to get the initial energy independant of ε when f ε and g ε satisfy somme good properties (namely hypothesis 1.2 detailed below).

Remark 1.1. In the following, we assume that L i = +∞. In fact, for t small enough, solution obtained on a finite domain with homogeneous boundary conditions is the same as solution obtained on a semi-finite domain. But effects of the junction does not depend on the choice of L i , neither on conditions eventually put at s i = L i .

The 1D limit problem

When ε tends to 0, the domain Ω ε "degenerates" into a "1D domain", namely the union of the n segments S i (left part of the figure 2). Intuitively, one expects, assuming f ε and g ε have good properties (with respect to ε), that the solution u ε "converges" to a "1D function" (in space), namely a function of time and the arclength s i along S i , solution of a "1D problem". It remains to give a more precise mathematical meaning to such a statement. To describe the "limit problem" inside the slots, we will use local normalized tangential and normal coordinates (s i , ν i ), that express that Ω ε i is isomorphic to the rectangle

Ω i =]0, +∞[× - c i 2 , c i 2 through the map x → (s i , ν i ) = (x • t i , x • n i /ε) from Ω ε i into Ω i
The reader can note that domains Ω i overlap (see for instance figure 3). Before writing our 1D limit problem, we consider additional hypothesis on f ε and g ε , namely:

Hypothesis 1.2. There exists ε 0 such that, for 0 < ε < ε 0 , for v ε = f ε or v ε = g ε , • supp v ε ∩ J ε = ∅, • on each Ω ε i , v ε (s, ν i ) = v ε (s) (called in the following v ε i ),
• v ε converges (in an appropriate space) to a function v 0 (which is a 1D function thanks to the two previous points). Under the hypothesis, let us define the following problem: find

INRIA

(u 0 i ) 1 i N such that                ∂ 2 u 0 i ∂t 2 (t, s i ) - ∂ 2 u 0 i ∂s 2 i (t, s i ) = 0 for (t, s i ) ∈ R * + ×]0, +∞[ u 0 i (0, s i ) = f 0 i for s i ∈]0, +∞[ ∂u 0 i ∂t (0, s i ) = g 0 i for s i ∈]0, +∞[ (6)
and

u 0 j (t, 0) = u 0 k (t, 0) for j = k and j c j ∂u 0 j ∂s j (t, 0) = 0 (7)
The system (6) can be seen as the formal limit of the system (1) when ε tends to 0. The coupling conditions [START_REF] Rubinstein | Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum[END_REF], known also in the literature as the Kirchhoff conditions, link the functions at the node of the graph. Intuition says that, on any subdomain in the slots, error between the exact solution and the approximated solution is in O(ε). More precisely, one has the following proposition:

Proposition 1.1. Given (δ i ) 1 i N ,

one can define:

• the sets Ω ε i,δ given as (see for instance figure 4):

Ω ε i,δ = {x ∈ Ω ε i such that x • t i δ i }
• the 2D function u 0 i defined as

u 0 i (t, x) = u 0 i (t, x • t i ) for t ∈ R + and x ∈ Ω ε i,δ RR n°7265 εσ 1 εσ 3 εσ 2 Figure 4: Configuration of the sets Ω ε i,δ
Then there exist some 1D function (

u 1 i ) 1 i N in space such that N i=1 1 ε Ω ε i,δ   ∂(u ε -u 0 i ) ∂t 2 + ∇(u ε -u 0 i ) 2   =   N i=1 +∞ δi   ε ∂ u 1 i ∂t 2 + ε∇ u 1 i 2     + O(ε 4 ) (8)
Proof. The proof of this proposition is a direct consequence of corollary 3.8 taking k = 1.

Remark 1.3. We can find Cauchy data such that u 1 i = 0 (for example, we choose Cauchy data such that the solution is a single wave contained in the first slot and going to infinity -this solution does not see the junction and does not have any interest for us). But one can see that when the junction is "excited", one has u 1 i = 0 for any later time.

An improved 1D approximate model

To describe our improved problem, we need to introduce some additional notations for our problem.

Additional information about the geometry

Definition 1.4 (Definition of σ i ). We introduce σ i , for 1 i N , as the smallest value for which (see figure 5 for an example)

{x ∈ Ω ε i such that x • t i εσ i } ∩   J ε ∪ j =i Ω ε j   = ∅
One can easily see that σ i does not depend of ε. From the definition of σ i , we add additional sets: we define Ω ε i,ext as (see for instance figure 6)

INRIA Γ 1 Γ 2 Γ 3 σ 1 σ 2 σ 3
Ω ε i,ext = {x ∈ Ω ε i / x • t i > εσ i }
and we define our new junction J ε as Note that we also have the decomposition

J ε = J ε ∪ N i=1 {x ∈ Ω ε / x • t i < εσ i } 000000000000
J ε Ω ε 3,ext Ω ε 1,ext Ω ε 2,ext
Ω ε = Ω ε 1,ext ∪ Ω ε 2,ext ∪ • • • ∪ Ω ε N,ext ∪ J ε
with domains that do not intersect each others. We keep J ε = ε J, where J is a fixed domain (i.e. not depending of ε) of R 2 . On domain J, we also define

• Γ i , for 1 i N (see again figure 5) as

Γ i = x ∈ ∂ J / x • t i = σ i and |x • n i | < c i 2 ,
• Γ neu as the remaining boundary (if exists).

Remark 1.5. One can easily imagine situations where boundary Γ N does not exist (for instance, a junction of three slots of same width with an angle of 2π/3 between two slots: the junction is an equilateral triangle whose edges can be identified to some Γ i ).

Solution of auxiliary problems and DtN maps

To allow writing of Improved Kirchhoff conditions, one uses the solution of N -1 auxiliary problems. Let us introduce, once Σ i has been identified to the segment ]0, c i [, T i as the non-local DtN operator defined as

T i : H 1 2 (]0, c i [) → H -1 2 (]0, c i [) ϕ → T i ϕ = ∞ p=1 pπ c i ci 0 ϕ( ν ′ )w i,p ( ν ′ )d ν ′ w i,p (9)
where w i,p is an orthonormal family of L 2 (]0, c i [) given by

w i,0 ( ν) = 1 c i , w i,p ( ν) = 2 c i cos pπ ν c i ( 10
)
Proposition 1.2. We have the following properties

• T i ∈ L H 1 2 (]0, c i [), H -1 2 (]0, c i [) • for any ϕ ∈ H 1 2 (]0, c i [), one has < T i ϕ, ϕ > -1 2 , 1 2 0
Proof. The proof is detailed in the appendix A.1.

Once we defined our DtN operators, we can define the following problem: find

W i ∈ H 1 ( J), for 1 i N -1, such that                    ∆W i = 0 in J c i (∇W i • n + T i W i ) = 1 on Γ i c i+1 (∇W i • n + T i+1 W i ) = -1 on Γ i+1 c k (∇W i • n + T k W i ) = 0 on Γ k , k = {i, i + 1} ∇W i • n = 0 on Γ neu (11)
with the additional condition

b J W i = 0 (12)
The additional condition (12) has been added to claim the following proposition (whose proof is detailed in appendix A.2):

INRIA Proposition 1.3. There exists a unique W i in H 1 ( J) satisfying (11) and (12).

We can then compute the matrix K ∈ M N -1 (R) given by

K i,j = 1 c j Γj W i - 1 c j+1 Γj+1 W i (13)
Another useful property is given by the following proposition (whose proof is detailed in appendix A.3) Proposition 1.4. K is a symmetric definite positive matrix.

Build the approximate problem

Thanks to sections 1.3.1 and 1.3.2, we can now give our improved Kirchhoff conditions. First, we introduce, for a family of functions

v i ∈ C 1 (R * + , H 1 (S i )) 1 i N
, with S i is the segment defined in section 1, the following vectors:

V ε (t) = (v j (t, εσ j )) 1 j N ∈ R N (14-i) ∂ S V ε (t) = (∂ sj v j (t, εσ j )) 1 j N ∈ R N (14-ii)
They represent the value of the i-th function and its normal derivate at the point on the boundary εΓ i (the blue circled point on figure 7).

εσ 1 εσ 3 εσ 2

Figure 7: Points (blue circles) used for the definition of the vectors (14-i) and (14-ii).

One can see that this definition will remain valid even if we consider instead a family of functions

v ε i ∈ C 1 (R * + , H 1 (εσ i , +∞)) 1 i N indexed by ε.
Let us now define the "jump matrix" J , the "average matrix" A and the "weight matrix" C by the following formulas:

• A is the N × N matrix given by (denoting | J| the area of J):

A = | J| N 2    1 . . . 1 . . . . . . 1 . . . 1    (15)
RR n°7265

• J is the N × N matrix given by

J = P T K -1 P ( 16
)
with K is the previously defined matrix and P is defined by

P =        1 -1 0 . . . . . . 0 0 1 -1 0 0 . . . . . . . . . . . . . . . . . . 0 0 1 -1 0 0 . . . . . . 0 1 -1        (17)
• C is a diagonal matrix of M N (R), given by

C =          c 1 0 . . . . . . 0 0 c 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . c N -1 0 0 . . . . . . 0 c N         
The Improved Kirchhoff model that we propose is the following: find

(ũ ε i ) 1 i N such that                ∂ 2 ũε i ∂t 2 (t, s i ) - ∂ 2 ũε i ∂s 2 i (t, s i ) = 0 for (t, s i ) ∈ R * + ×]εσ i , +∞[ ũε i (0, s i ) = f ε i for s i ∈]εσ i , +∞[ ∂ ũε i ∂t (0, s i ) = g ε i for s i ∈]εσ i , +∞[ (18)
and

C∂ S Ũ ε (t) = 1 ε J + εA ∂ 2 ∂t 2 Ũ ε (t), for t ∈ R * + ( 19
)
Remark 1.6. It is clear that, when ε tends to 0, (18) gives [START_REF] Kuchment | Graph models for waves in thin structures[END_REF]. The not so clear point is that, when ε tends to 0, (19) gives [START_REF] Rubinstein | Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum[END_REF]. However, at least formally, Ũ ε (t) tends to Ũ 0 (t), and the formal limit of (19) multiplied by ε leads to the fact that Ũ 0 (t) is in the kernel of J , which is equal to Vect{(1, . . . , 1) t } This gives the left part of [START_REF] Rubinstein | Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum[END_REF]. Then, going back to (19) and doing the scalar product with (1, . . . , 1) t leads to

j c j ∂u ε j ∂s j (t, εσ j ) = ε | J| N j ∂ 2 u j ∂t 2 (t, εσ j )
The formal limit of this relation gives the right part of [START_REF] Rubinstein | Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum[END_REF].

INRIA

For the Improved Kirchhoff problem (18, 19), one may define the approximate energy

E ε (t, v) = N i=1 c i 2 +∞ εσi ∂v ∂s i (t, s ′ ) 2 + ∂v ∂t (t, s ′ ) 2 ds ′ + 1 2ε (J V ε (t)) • V ε (t) + ε 2 A ∂V ε ∂t (t) • ∂V ε ∂t (t) (20)
Proposition 1.5. We have the following energy conversation relation

E ε (t, ũε ) = E ε (0, ũε ) = N i=1 c i 2 +∞ εσi ∂f ε ∂s i (s ′ ) 2 + |g ε (s ′ )| 2 ds ′ (21)
This proposition will be proved in section 4.2.

One idea, of course, is to compare the difference between the solution of the exact problem (1) and the solution of the 1D Improved problem (18, 19). We compare this in four steps:

• we estimate the difference between the solution of the exact problem and an expansion in powers of ε,

• we estimate the difference between the solution of the 1D Improved problem and another expansion in powers of ε,

• we remark in section 4.2 that the two expansions are the same one for at least the two first terms,

• we use some classical triangular inequality to conclude.

Finally, one gets the following proposition (which is an improvement of proposition 1.1):

Proposition 1.6. Given (δ i ) 1 i N , one can define:

• the set Ω ε i,δ given as (see for instance figure 4):

Ω ε i,δ = {x ∈ Ω ε i such that x • t i δ i }
• the 2D function u ε i defined as

u ε i (t, x) = ũε i (t, x • t i ) for t ∈ R + and x ∈ Ω ε i,δ
Then if f ε i ∈ H 5 (S i ) and g ε i ∈ H 4 (S i ), for 1 i N (see back hypothesis 1.2), there exist some 1D functions (u i ) 1 i N in space such that:

N i=1 1 ε Ω ε i,δ   ∂(u ε i -u ε i ) ∂t (t, x) 2 + ∇(u ε i -u ε i )(t, x) 2   dx = N i=1 +∞ δi ε 2 ∂u i ∂t (t, s i ) 2 + ε 2 ∇u i (t, s i ) 2 ds i + O(ε 6 ) (22)

RR n°7265

Proof. For this proof, we denote by E δ the following quantity:

E ε δ (v) = N i=1 1 ε Ω ε i,δ ∂v ∂t (t, x) 2 + |∇v(t, x)| 2 dx
Thanks to the hypothesis on the Cauchy data, we can bring the third order approximation for both u ε i and u ε i . We denote then by u ε i,2 the approximation of u ε i to second order, and u ε i,2 the approximation of u ε i to second order. Then one has that

E ε δ (u ε i -u ε i ) -E ε δ (u ε i,2 -u ε i,2 ) E ε δ (u ε i -u ε i,2 ) + E ε δ ( u ε i * u ε i,2 ) (23)
Now, for the left member size, since u 0 i = u 0 i and u 1 i = u 1 i (this point will be proved later in section 4.2), there exist a function u i such that

u ε i,2 -u ε i,2 = ε 2 u i
One can easily see that u i is 1D in space and does not depend on ε. To conclude, we use both corollaries 3.8 and 4.1 for n = 2 to treat the right member of (23), and we obtain the desired result.

INRIA 2 The formal expansions

As we said in the introduction, as the problem is multi-scale (in space), it is not possible to write a uniform expansion for the solution everywhere in the domain Ω ε . The method of matched asymptotics will lead us, we have to consider N + 1 distinct zones, respectively N slot zones and a junction zone, in which different expansions will be obtained. However, contrarily to the naive intuition, this domain decomposition does not correspond to the partition (2) of Ω ε : in the method of matched asymptotics, the different domains must overlap, the idea being that the different expansions must "coincide" in the overlapping zones.

An overlapping domain decomposition

In the following, we will denote by C the class of positive continuous functions of ε > 0 that tend to 0 when ε → 0, less rapidly than ε| log(ε)| (a typical example is ε β , with β strictly less than 1). Given ϕ in C, we define, for 1 i N , the i-th slot zone by (see figure 8):

C = ϕ : R * + → R * + / lim ε→0 ϕ(ε) = 0 and lim ε→0 ϕ(ε) ε| log(ε)| = +∞ (24)
σ 1 ε σ 2 ε σ 3 ε 2ϕ(ε) ϕ(ε) σ 1 ε σ 2 ε σ 3 ε 2ϕ(ε) ϕ(ε)
Ω ε i (ε) = {x ∈ Ω ε i / ϕ(ε) x • t i } (⊂ Ω ε i )
The junction zone is defined by (see again figure 8):

J ε (ε) = ε J ∪ N i=1 {x ∈ Ω ε i / 0 x • t i < 2ϕ(ε)} (contains J ε )
in such a way that we have

Ω ε = Ω ε 1 (ε) ∪ Ω ε 2 (ε) ∪ • • • ∪ Ω ε N (ε) ∪ J ε (ε)
with N overlapping regions (see figure 9)

O i (ε) = Ω ε i (ε) ∩ J ε (ε) = (x, y) ∈ Ω ε / ϕ(ε) < x • t i < 2ϕ(ε) and - c i ε 2 < x • n i < c i ε 2
The mapping

x → (x • t i , x • n i /ε) (25) σ 1 ε σ 2 ε σ 3 ε 2ϕ(ε) ϕ(ε)
Ω ε i (ε) into the rectangle Ω i (ε)
with

Ω i (ε) =]ϕ(ε), +∞[× - c i 2 , c i 2
Note that Ω i (ε) increases when ε decreases and converges to Ω i when ε tends to 0. In the same way, x → x = x/ε (26) maps the domain J ε (ε) onto J(ε), a domain which increases when ε decreases and converges to the unbounded domain:

J = J ∪ N i=1 B i , B i = { x / x • t i 0, | x • n i | < c i /2}

Local expansions and basic equations

We formulate our Ansatz for the asymptotics expansions which consists, in each zone after scaling ((25) or (26)), in looking for power series expansions with respect to ε1 . In other words, we look for functions

u k i : Ω i → C and U k : J → C, k ∈ N,
such that, at least formally

u ε (t, s i , ε ν i ) = ∞ k=0 ε k u k i (t, s i , ν i ) + o(ε ∞ ), in R * + × Ω i (ε) (27) u ε (t, ε x) = ∞ k=0 ε k U k (t, x) + o(ε ∞ ), in R * + × J(ε) (28)
It remains to obtain the equations that will determine the functions u k i and U k . For the u k i 's, we substitute formally the expansions (27) in the 2D acoustic equation written in R * + × Ω i (ε), using the scaled coordinates (s, ν), and we identify the terms with the same power of ε. Straightforward manipulations lead to:

∂ 2 u 0 i ∂ ν 2 i = 0, ∂ 2 u 1 i ∂ ν 2 i = 0, ∂ 2 u k i ∂t 2 - ∂ 2 u k i ∂s 2 i - ∂ 2 u k+2 i ∂ ν 2 i = 0, k 0 (29)
with the Neumann boundary condition along the "lateral" sides

∂u k i ∂ ν i t, s i , - c i 2 = ∂u k i ∂ ν i t, s i , c i 2 = 0, t ∈ R * + , s i ∈]0, +∞[, k 0 (30)
From ( 29) and (30), we deduce that Proposition 2.1. For 1 i N , for k ∈ N, one has

u k i (t, s i , ν i ) = u k i (t, s i ), k 0 (31-i) ∂ 2 u k i ∂t 2 - ∂ 2 u k i ∂s 2 i = 0, k 0 (31-ii)
Proof. We prove (31-i) and (31-ii) for a given i by induction on k. For k = 0, 1, the first two equations of (29) combined with (30) show that u 0 i and u 1 i are independent of ν. Then, integrating the third equation of (29) written for k = 0 (respectively k = 1) with respect to ν and using the boundary conditions (30) for k = 2 (respectively for k = 3), we see that u 0 i (respectively u 1 i ) satisfies (31-ii). Assume that (31-i) and (31-ii) holds up to k = p. Then, the third equation of (29) written for k = p -1 combined with (30) show that u p+1 i is independent of ν. Next, integrating the third equation of (29) written for k = p+1 with respect to ν and using the boundary conditions (30) leads to (31-ii) for k = p + 1.

Remark 2.1. Now we may explain at least formally why the second line of hypothesis 1.2 is almost necessary. The proposition 2.1 tell us that each term of the asymptotic expansion (27) does not depend on ν i . Now assume that there exists a family of function

(f k i ) k∈N defined on Ω i such that f ε (s i , ε ν i ) = ∞ k=0 ε k f k i (s i , ν i ) + o(ε ∞ ), in Ω i (ε)
We can see that

u k i (0, •) = f k i .
Since u k i does not depend on ν and is continuous with respect to t, one can deduce that f k i should neither depends on ν. In the same way, we get the same result for g k i . Moreover, from Cauchy data in (1), we deduce

u k i (0, •) = f k i , 1 i N, k 0 (32-i) ∂u k i ∂t (0, •) = g k i , 1 i N, k 0 (32-ii)
To obtain the equations for the U k 's, we substitute formally the expansion (28) in the 2D acoustic equation written in J(ε), using the scaled coordinates x, and we identify the terms with the same power of ε. This permits us to see that the U k 's satisfy embedded Laplace's equations

∆U 0 = 0, ∆U 1 = 0, ∆U k+2 = ∂ 2 U k ∂t 2 , k ∈ N, in R * + × J (33)
with Neumann boundary conditions

∂U k ∂ n = 0, on ∂ J, k ∈ N ( 34
)
Remark 2.2. In the sequel, we shall adopt the convention that all quantities super-indexed by k (such as U k , u k i , . . . ) are 0 for negative values of k. This will be useful to simplify some formulas. For instance, with this convention the last equation of ( 33) is also valid for k = 0, 1.

Matching conditions

Equations (31) to (34) are not sufficient to characterize the functions (u k i , U k )'s: we miss boundary conditions at s = 0 for the u k i 's and additional conditions at infinity for the U k 's. These conditions, namely the matching conditions, will couple the u k i 's and the U k 's. To derive them, one writes that the two expansions ( 27) and (28) must coincide in each overlapping zone O i (ε). Introducing the canonical semi-strip B i defined by

B i = x ∈ B i / x • t i > σ i
(see the definition of σ i given by 1.4)

and calling U k i the restriction of U k on the i-th canonical semi-strip let us allow to express the following matching conditions:

+∞ k=0 ε k U k i (t, s i /ε, ν i /ε) + o(ε ∞ ) = +∞ k=0 ε k u k i (t, s i , ν i /ε) + o(ε ∞ ) in R * + × O i (ε) (35)
To express more precisely these matching conditions, it is useful to describe the form of the functions U k i 's in the semi-strips B i : this is the object of the next section.

Modal expansion of solutions of embedded Laplace equations

Let consider, for a given i, U k i :

B i → R satisfying ∆U 0 i = 0, ∆U 1 i = 0, ∆U k+2 i = ∂ 2 U k i ∂t 2 , in B i , ∂U k i ∂ ν i = 0 for | ν i | = c i /2
(36) Later, in section 2.3.2, the results of the present section will be applied to the restriction of the U k 's on B i , where U k 's are the coefficients of the expansion (28).

In B i , we shall use the local coordinates

( s i , ν i ) = ( x • t i , x • n i ) such that x ∈ B i ⇐⇒ ( s i , ν i ) ∈]σ i , +∞[×] -c i /2, c i /2[

INRIA

The behaviour of the fields U k i in the semi-strips B i is easily described by using separation of variables in ( s i , ν i ) coordinates, which introduce naturally the orthonormal family w i,p defined by

w i,0 ( ν i ) = 1 c i , w i,p ( ν i ) = 2 c i cos pπ ν i c i + pπ 2 (37)
Remark 2.3. One can easily see that

w i,p ( ν i ) = w i,p ( ν i + c i /2)
where w i,p is the family defined by [START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF]. The "basis" w i,p is adapted to the Neumann conditions at ν i = ±c i /2, and there exists 1D functions (in space)

U k i,p (t, s i ) such that U k i (t, s i , ν i ) = p∈N U k i,p (t, s i ) w i,p ( ν i ) (38)
with (thanks to the fact that w i,p is orthonormal):

U k i,p (t, s i ) = ci/2 -ci/2 U k i (t, s i , ν ′ i ) w i,p ( ν ′ i )d ν ′ i
If we substitute formally the expression (38) into the equations (36), we obtain

∀k 1, ∀p ∈ N, ∂ 2 U k i,p ∂ s 2 i (t, s i ) - pπ c i 2 U k i,p (t, s i ) = 0 (39-i) ∀k 2, ∀p ∈ N, ∂ 2 U k i,p ∂ s 2 i (t, s i ) - pπ c i 2 U k i,p (t, s i ) = ∂ 2 U k-2 i,p ∂t 2 (t, s i ) (39-ii)
The resolution of (39) is a tedious but quite simple exercise on ordinary differential equations (the idea is to solve this system with respect to s i on a given time t fixed). In what follow, we reproduce some results of [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF], that we present in a slightly different form, more adapted to the purpose of this report. After having remarked that the change of unknowns

U k i,p (t, s i ) = exp(± pπ c i s i )V k i,p,± (t, s i )
leads to the equations (with the convention of the remark 2.2)

∂ 2 V k i,p,± ∂ s 2 i (t, s i ) ± 2 pπ c i ∂V k i,p,± ∂ s i (t, s i ) = ∂ 2 V k-2 i,p,± ∂t 2 (t, s i ) (40)
One can see, for k = 0, 1, that the equation (40) does not depend on time.

We can write V k as sum of products of functions depending on t and functions depending on s i . The idea is to work with separation of variables and to search

V k i,p,± , for k ∈ N, under the form V k i,p,± (t, s i ) = φ k i,p,± ( s i )ψ k i,p,± (t)
, and one can see that the family φ has to satisfy the following equation (for this part, we don't care about dependance in time for functions V k i,p,± ):

∂ 2 φ k i,p,± ∂ s 2 i ( s i ) ± 2 pπ c i ∂φ k i,p,± ∂ s i ( s i ) = C k i,p,± φ k-2 i,p,± ( s i ) (41)
Remark 2.4. One can see later (in the proof of lemma 2.3) that is it judicious to take C k i,p,± = 1, for each quartet (i, p, k, ±). In [START_REF] Joly | Construction and analysis of improved kirchhoff conditions for acoustic wave propagation in a junction of thin slots[END_REF] we took C k i,p,± = -1, this coefficient being dictated by the coefficient of the right member of embedded Laplace equations.

We introduce, for each p ∈ N, two sequences of polynomial solutions (in space) of ( 41)

(n k i,p , d k i,p
), k ∈ N, which are defined inductively on k, for each p ∈ N and are identically 0 for odd values of k.

• The value p = 0 plays a particular role, since equation ( 41) degenerates (the equation with "± = +" is the same as the equation with "± = -").

For k = 0, 1 one has

n 0 i,0 ( s i ) = s i , n 1 i,0 ( s i ) = 0, d 0 i,0 ( s i ) = 1, d 1 i,0 ( s i ) = 0 (42) continuing for k 2 with ∂ 2 n k i,0 ∂ s 2 i = n k-2 i,0 , n k i,0 (0) = ∂n k i,0 ∂ s i (0) = 0 (43-i) ∂ 2 d k i,0 ∂ s 2 i = d k-2 i,0 , d k i,0 (0) = ∂d k i,0 ∂ s i (0) = 0 (43-ii)
It is easy to see that, for even k, one recovers the monomials of the series expansion of exp( s i ):

n 2m i,0 ( s i ) = s 2m+1 i (2m + 1)! , d 2m i,0 ( s i ) = s 2m i (2m)! , (44)
• For p 1, one starts from

n 0 i,p ( s i ) = 1, n 1 i,p ( s i ) = 0, d 0 i,p ( s i ) = 1, d 1 i,p ( s i ) = 0, for p 1 (45)
Then, (n k i,p , d k i,p ) are defined as the polynomial solutions of

∂ 2 n k i,p ∂ s 2 i + 2pπ c i ∂n k i,p ∂ s i = n k-2 i,p , n k i,p (0) = 0 (46-i) ∂ 2 d k i,p ∂ s 2 i - 2pπ c i ∂d k i,p ∂ s i = d k-2 i,p , d k i,p (0) = 0 (46-ii)
Note the difference of sign between (46-i) and (46-ii) for the first derivative term.

The following proposition gives some properties about the functions

(n k i,p , d k i,p ): Proposition 2.2. Let (n k i,p , d k i,p
) k∈N,p 1 be a polynomial family of functions satisfying (45) and (46), then this family is well-defined and is unique,

-(n 2m i,p , d 2m i,p ) have degree m, -n 2m+1 i,p = d 2m+1 i,p = 0, INRIA -we have a relation between n 2m
i,p and n 2m i,p given by:

n 2m i,p ( s i ) = d 2m i,p (-s i )
we have a relation between n 2m i,p and n 2m i,1 given by:

n 2m i,p ( s i ) = p -k n 2m i,1 (p s i )
Proof. The proof of this proposition is very similar to the proof of the proposition 2.3 of [START_REF] Joly | Propagation of an acoustic wave in a junction of two thin slots[END_REF].

Remark 2.5. The fact that the functions n and d are identically 0 for odd values of k is due to the fact that the equations satisfied by

U 2m+1 i,p
, for m ∈ N, are identical to the equations satisfied by U 2m i,p (this is a choice of notation that permit us the writing of future lemmas, regardless whether k is odd or even).

Next, we construct two families of functions n k i,p and

d k i,p from B i into R, for (k, p) ∈ N 2 , by n k i,p ( x) = exp(pπ s i /c i )n k i,p ( s i ) w i,p ( ν i ) (47-i) d k i,p ( x) = exp(-pπ s i /c i )d k i,p ( s i ) w i,p ( ν i ) (47-ii)
that constitute particular families of embedded Laplace's equations:

∀p ∈ N, ∀k ∈ N, ∆n k+2 i,p = n k i,p , ∆d k+2 i,p = d k i,p , in B i (48)
with homogeneous Neumann boundary conditions at

ν i = ±c i /2.
The families (n k i,p , d k i,p ) allow us to express the following lemma Lemma 2.3 (Fundamental lemma for the expansion of U k i ). Let (U k i ) k∈N be a family of functions satisfying (36), then there exists two sequences (η k i,p , δ k i,p ) (k,p)∈N 2 of complex functions depending only on t such that:

U k i (t, x) = p∈N k l=0 ∂ l η k-l i,p ∂t l (t)n l i,p ( x) + ∂ l δ k-l i,p ∂t l (t)d l i,p ( x) in R * + × B i (49)
Proof. We will prove the relation (49) by induction on k.

• Initialization: for k = 0, 1 and for a given t ∈ R * + , one can see that ∆U k 0 (t, •) vanishes. Hence, there exists two sequences of complex numbers, that we call η k i,p (t) and δ k i,p (t), such that

U k i (t, x) = p∈N η k i,p (t)n 0 i,p ( x) + δ k i,p (t)d 0 i,p ( x) in R * + × B i (50)
For k = 0, the expressions (50) and (49) are the same ones. For k = 1, since n 1 i,p = d 1 i,p = 0 for all p ∈ N, one can see also that the expressions (50) and (49) remain the same ones.

• Heredity: let us admit that the sequences (η k i,p , δ k i,p ) have been constructed up to l = k -1. Then let us introduce (remember that n 1 i,p = d 1 i,p = 0):

U k i (t, x) = p∈N k l=1 ∂ l η k-l i,p ∂t l (t)n l i,p ( x) + ∂ l δ k-l i,p ∂t l (t)d l i,p ( x) = p∈N k l=2 ∂ l η k-l i,p ∂t l (t)n l i,p ( x) + ∂ l δ k-l i,p ∂t l (t)d l i,p ( x)
We have (successively, we use (48), apply the change of index l → l -2, and use (49

) for k -2): ∆ U k i (t, x) = p∈N k l=2 ∂ l η k-l i,p ∂t l (t)∆n l i,p ( x) + ∂ l δ k-l i,p ∂t l (t)∆d l i,p ( x) = p∈N k l=2 ∂ l η k-l i,p ∂t l (t)n l-2 i,p ( x) + ∂ l δ k-l i,p ∂t l (t)d l-2 i,p ( x) = p∈N k-2 l=0 ∂ l+2 η k-2-l i,p ∂t l+2 (t)n l i,p ( x) + ∂ l+2 δ k-2-l i,p ∂t l+2 (t)d l i,p ( x) = ∂ 2 U k-2 i ∂t 2 (t, x)
The function

U k i (t, •) -U k i (t,
•) being harmonic in B i , we know there exist two sequences complex numbers that we choose to denote respectively by (η k i,p (t)) p∈N and (δ k i,p (t)) p∈N , such that

U k i (t, x) -U k i (t, x) = p∈N η k i,p (t)n 0 i,p ( x) + δ k i,p (t)d 0 i,p ( x)
and the proof is complete.

From the previous lemma, it is natural to introduce the (increasing) spaces

V k i = span (n l i,p , d l i,p ), p ∈ N, l k , V i = k∈N V k i
and let us introduce, for the functions U ∈ V 0 i , the two natural families of linear forms N i,p and D i,p defined such that:

∀U ∈ V 0 i , U ( x) = ∞ p=0 N i,p (U )n 0 i,p (x) + D i,p (U )d 0 i,p (x)
Remark 2.6. One can see these linear forms as "traces" of the function on s i = 0 (although s i = 0 does not belong to B i ).

Another way to write lemma 2.3 is Corollary 2.4. Let (U k i ) k∈N be a family of functions satisfying (36), then:

• U k i (t, •) ∈ V k i
• It can be written as

U k i (t, x) = p∈N k l=0 ∂ l N i,p (U k-l i (t, •)) ∂t l (t)n l i,p ( x) + ∂ l D i,p (U k-l i (t, •)) ∂t l (t)d l i,p ( x) INRIA 2.

Derivation of the matching conditions

To express the matching conditions, we simply write that the two expansions ( 27) and (28) must coincide in the overlapping zones O i (ε) (see back figure 9 page 16). In such zones, we have the following relations (we recall here (35) by denoting

U k i the restriction of U k to B i ) +∞ k=0 ε k U k i (t, s i /ε, ν i /ε) + o(ε ∞ ) = u ε (t, s i , ν i ) = +∞ k=0 ε k u k i (t, s i , ν i /ε) + o(ε ∞ ) for t ∈ R, ϕ(ε) < s i < 2ϕ(ε) and |ν i | < εc i /2.
We denote R the right equality of this relation and L the left one.

Treatment of R This is the simplest one. One has to consider that each function u k i does not depend on the third variable ν i and the fact that φ belongs to C defined in (24), which implies that s i tends to 0 as ε tends to 0. By using a Taylor development on s i = 0 for each function u k i , one has

u ε (t, s i , ν i ) = k∈N m∈N 1 m! ∂ m u k i ∂s m i (t, 0) ε k s m i + o(ε ∞ ) (51)
Treatment of L Here one has more work to do. First, we use the corollary 2.4 of the fundamental lemma to express each U k i on the family (n m i,p , d m i,p ) (p,m)∈N 2 defined in section 2.3.1, and we get

u ε (t, s i , ν i ) = k∈N ε k p∈N k m=0 ∂ m N i,p (U k-m (t, •)) ∂t m (t)n m i,p s i ε , ν i ε + ∂ m D i,p (U k-m (t, •)) ∂t m (t)d m i,p s i ε , ν i ε + o(ε ∞ )
that can be rewritten as (summing for m k is like summing for k m, for convergent series)

u ε (t, s i , ν i ) = m∈N k m p∈N ε k ∂ m N i,p (U k-m (t, •)) ∂t m (t)n m i,p s i ε , ν i ε + ∂ m D i,p (U k-m (t, •)) ∂t m (t)d m i,p s i ε , ν i ε + o(ε ∞ )
Since the function ϕ belongs to C, and since the functions d m i,p are exponentially decreasing at s i = +∞, the corresponding terms in the previous sum can be "put into" the o(ε ∞ ) part. For the rest of the sum, we distinguish the terms for p = 0, for which we use the formulas (44) and (47), from the terms corresponding to p 1 (which are exponentially increasing at infinity):

u ε (t, s i , ν i ) = 1 c i m∈N k 2m ε k ∂ 2m N i,0 (U k-2m (t, •)) ∂t 2m (t) 1 (2m + 1)! s i ε 2m+1 + 1 c i m∈N k 2m ε k ∂ 2m D i,0 (U k-2m (t, •)) ∂t 2m (t) 1 (2m)! s i ε 2m + p 1 m∈N k m ∂ m N i,p (U k-m (t, •)) ∂t m (t)n m i,p s i ε exp pπs i εc i w i,p ν i ε
Using the change of index k → k+2m+1 in the first line (respectively k → k+2m in the second line) of the previous equation, one gets

u ε (t, s i , ν i ) = 1 c i m∈N k -1 ∂ 2m N i,0 (U k+1 (t, •)) ∂t 2m (t) 1 (2m + 1)! ε k s 2m+1 i + 1 c i m∈N k∈N ∂ 2m D i,0 (U k (t, •)) ∂t 2m (t) 1 (2m)! ε k s m i + p 1 m∈N k m S p,m,n (ε, s i , ν i ) (52)
with

S p,m,n (ε, s i , ν i ) = ∂ m N i,p (U k-m (t, •)) ∂t m (t)n m i,p s i ε exp pπs i εc i w i,p ν i ε
Conclusion Finally, the formal identification of the expressions (51) and (52) in the overlapping zone O i (ε), as functions of s i and ε, will lead us to our matching conditions.

First, for p 1, after multiplication of (51) and (52) by w i,p νi ε and integration over ν i , we get

m∈N n m i,p s i ε   k m ∂ m N i,p (U k-m (t, •)) ∂t m (t)   = 0
The proposition 2.2 implies that the functions n m i,p are linearly independent and one deduces from that N i,p (U k-2m (t, •)) = 0, ∀m 0, ∀k 2m, that is to say:

N i,p (U k (t, •)) = 0, p 1, k 0, 1 i N (53)
which express the absence of exponentially growing terms. We can see that (53) leads to the following condition U k grows as most polynomially at infinity in J (54)

INRIA

Next, it remains to identify power series expansions.The identification of the terms in ε k s m i , distinguishing even and odd values of m, leads to

∂ 2m D i,0 (U k (t, •)) ∂t 2m (t) = √ c i ∂ 2m u k i ∂s 2m i (t, 0) ∂ 2m N i,0 (U k (t, •)) ∂t 2m (t) = √ c i ∂ 2m+1 u k-1 i ∂s 2m+1 i (t, 0)
Using the fact that each u k i solves the 1D time wave equation, we have

∂ 2m u k i ∂s 2m i (t, 0) = ∂ 2m u k i ∂t 2m (t, 0)
and we get the "Dirichlet" and "Neumann" matching conditions, namely:

D i,0 (U k (t, •)) = √ c i u k i (t, 0), ∀k ∈ N, ∀t ∈ R * + (55-i) N i,0 (U k (t, •)) = √ c i ∂u k-1 i ∂s i (t, 0), ∀k ∈ N, ∀t ∈ R * + (55-ii)

Summary

Finally, one gets the coupled system of equations and matching conditions, that are (we recall here the equations (31), (32), (33) and (34), and the matching conditions (55)):

• the equations written on the slots zone:

u k i (t, s i , ν i ) = u k i (t, s i ), k 0 ∂ 2 u k i ∂t 2 - ∂ 2 u k i ∂s 2 i = 0, k 0 u k i (0, •) = f k i , 1 i N, k 0 ∂u k i ∂t (0, •) = g k i , 1 i N, k 0
• the equations written on the junction zone:

∆U 0 = 0, ∆U 1 = 0, ∆U k+2 = ∂ 2 U k ∂t 2 , k ∈ N, in R * + × J ∂U k ∂ n = 0, on ∂ J, k ∈ N
• the matching conditions between the slots zones and the junction zone:

D i,0 (U k (t, •)) = √ c i u k i (t, 0), ∀k ∈ N, ∀t ∈ R * + N i,0 (U k (t, •)) = √ c i ∂u k-1 i ∂s i (t, 0), ∀k ∈ N, ∀t ∈ R * + RR n°7265
3 Justification and error estimates

In this paragraph, our goal is to prove that the functions (u k i ) k∈N,1 i N and (U k ) k∈N are uniquely defined, and that there exists an approximate function built from these functions which differs from the solution of the exact problem with some power of ε that is increasing with the order of the approximation we consider.

Existence and uniqueness of the formal expansion

In this section, our goal is to prove that the equations (31), (32), ( 33) and (34), together with the matching conditions (55), define unique families (u k i ) k∈N,1 i N and (U k ) k∈N . To reach our goal, we first formulate an equivalent problem where the unknowns (U k ) k∈N are restricted to the junction J.

Restriction to a bounded domain of the problem for the U k

Our goal in this section is to characterize the restrictions of the functions U k 's to the junction J by giving exact Dirichlet to Neumann boundary conditions at the interfaces Γ i (see again figure 5).

Let us start with the corollary 2.4 of the fundamental lemma, applied to the restriction of U k to the semi-slot B i , and then apply (53), then we get, separating l = 0 from l 1:

U k i (t, x) = N i,0 (U k (t, •))n 0 i,0 ( x) + p∈N D i,p (U k (t, •))d 0 i,p ( x)
+ k l=1 ∂ l N i,0 (U k-l (t, •)) ∂t l n l i,0 ( x) + p∈N k l=1 ∂ l D i,p (U k-l (t, •)) ∂t l d l i,p ( x)
Remember that, for our junction zone J, the boundary Γ i is parametrized by

Γ i = x ∈ ∂ J / x • t i = σ i and |x • n i | < c i 2 ,
then, in the previous sum, we make the index change l → 2l (remember also that n l i,p and d l i,p vanish for odd values of l), and the use of analytically expressions of n l i,p and d l i,p , leads to

INRIA U k i (t, σ i , ν i ) = N i,0 (U k (t, •)) σ i √ c i + D i,0 (U k (t, •)) 1 √ c i + p 1 D i,p (U k (t, •)) exp - pπσ i c i w p,i ( ν i ) + 1 √ c i k/2 l=1 ∂ 2l N i,0 (U k-2l (t, •)) ∂t 2l σ 2l+1 i (2l + 1)! + 1 √ c i k/2 l=1 ∂ 2l D i,0 (U k-2l (t, •)) ∂t 2l σ 2l i (2l)! + k/2 l=1 p 1 ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l exp - pπσ i c i d 2l i,p (σ i ) w p,i ( ν i ) (56) From (56), since ( w p,i ) p∈N is an orthonormal family of L 2 (Γ i ), one can see that Γi U k i (t, σ i , ν i ) w p,i ( ν i )d ν i = D i,p (U k (t, •))e " - pπσ i c i " + k/2 l=1 ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l e " - pπσ i c i " d 2l i,p (σ i ) (57)
Moreover, on Γ i , the normal derivative of U k is the same as the derivative among s i . After some computations, one has

∂U k i ∂s i (t, σ i , ν i ) = N i,0 (U k (t, •)) 1 √ c i - p 1 pπ c i D i,p (U k (t, •))e " - pπσ i c i " w p,i ( ν i ) + 1 √ c i k/2 l=1 ∂ 2l N i,0 (U k-2l (t, •)) ∂t 2l σ 2l i (2l)! + 1 √ c i k/2 l=1 ∂ 2l D i,0 (U k-2l (t, •)) ∂t 2l σ 2l-1 i (2l -1)! - k/2 l=1 p 1 pπ c i ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l e " - pπσ i c i " d 2l i,p (σ i ) w p,i ( ν i ) + k/2 l=1 p 1 ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l e " - pπσ i c i " ∂d 2l i,p ∂s i (σ i ) w p,i ( ν i ) (58)
The most important point is that, using (57), the sum of the second and fifth lines of (58) is equal to

- p 1 pπ c i Γi U k i (t, σ i , ν i ) w p,i ( ν i )d ν i
We can now, by using (55-ii) and by using the definition of non-local DtN operator T i given by [START_REF] Semin | Numerical resolution of the wave equation on a network of slots[END_REF], sum up the previous information by writing the following relation: on Γ i , one has

∂U k ∂ n (t, •) + T i U k (t, •) = ∂u k-1 i ∂s i (t, 0) + k/2 l=1 ∂ 2l+1 u k-2l-1 i ∂t 2l ∂s i (t, 0) σ 2l i (2l)! + 1 √ c i k/2 l=1 ∂ 2l D i,0 (U k-2l (t, •)) ∂t 2l σ 2l-1 i (2l -1)! + k/2 l=1 S i,k,l (σ i ) (59)
with

S i,k,l (σ i ) = p 1 ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l exp - pπσ i c i ∂d 2l i,p ∂s i (σ i ) w p,i
For the right hand side of (59), one can see that we have two parts:

• one part linked to the knowledge of U k ′ , for k ′ < k,

• one more interesting part linked to the derivative of u k-1 i (t, s i ) at s i = 0.

Finally, we obtain a problem "equivalent" to ((31), (32), (33), (34), (55)) by replacing (55-ii) to the DtN condition (59). The precise statement is the following

(denoting C 1,2 (V ) = C 0 (R * + , H 1 (V )) ∩ C 1 (R * + , L 2 (V )): Theorem 3.1. Let (u k j ) ∈ C 1,2 (]0, L j [) 1 j n and U k ∈ C 1,2,loc (
J) be a solution of ((31), (32), (33), (34)) with the matching conditions (55), then

(u k 1 , . . . , u k n , U k ) ∈ C 1,2 (]0, L 1 [) × • • • × C 1,2 (]0, L N [) × C 1,2 ( J), k 0
is solution of ((31), (32)), ((33), (34)) b J , (55-i) and (59), where ((33), (34)) b J holds the restriction of ((33), (34)) respectively to J and

∂ J ∩ ∂ J. Reciprocally, if (u k j ) ∈ C 1,2 (]0, L j [) 1 j n and U k ∈ C 1,2,loc (
J) are solution of ((31), (32)), ((33), (34)) b J , (55-i) and (59), then by extending U k to each B i via

U k i (t, x) = √ c i ∂u k-1 i ∂s i (t, 0)n 0 i,0 ( x) + p∈N D i,p (U k (t, •))d 0 i,p ( x) + √ c i k l=1 ∂ l+1 u k-l-1 ∂t l ∂s i (t, 0)n l i,0 ( x) + p∈N k l=1 ∂ l D i,p (U k-l (t, •)) ∂t l d l i,p ( x) Then (u k 1 , . . . , u k n , U k ) ∈ C 1,2 (]0, L 1 [) × • • • × C 1,2 (]0, L N [) × C 1,2
,loc ( J), k 0 is a solution of( (31), (32), (33), (34)) with the matching conditions (55)

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Proof. The direct statement has been proved. For the reciprocal, let us consider

(u k 1 , . . . , u k n , U k ) ∈ C 1,2 (]0, L 1 [) × • • • × C 1,2 (]0, L N [) × C 1,2
,loc ( J), k 0 solution of ((31), (32)), ((33), (34)) b J , (55-i) and (59), and let us extend U k to each B i . It is quite easy to prove that U k satisfies embedded Laplace equations on B i (the proof of this point is the same as the proof of the fundamental lemma 2.3). By looking the modal expansion of the corollary and the modal expansion written above, (55-i) is immediate. A priori, the most difficult point is to show that Dirichlet and Neumann traces are continuous on Γ j ; however these conditions are satisfied thanks to (59) (this condition has been built to satisfy such things).

Auxiliary lemmas

Existence and uniqueness is done by induction on k. According to theorem 3.1, it suffices to consider the problem ((31), (32)), ((33), (34)) b J , (55-i) and (59). To clearly identify the recursion process, it is useful to reformulate this problem in a more decoupled way (we mean between u k i and U k , at each step k), which is also useful from the computational point of view.

To achieve such a decoupling the idea is first to consider (59) as a boundary condition for U k , next to formulate a 1D transmission problem for u k i . In this sense, we have to prove first the following two technical lemmas. Lemma 3.2. Given Φ ∈ L 2 ( J) and (g i ∈ H -1 2 (Γ i )) 1 i N , there exists U ∈ H 1 ( J), which is unique up to an additive constant, such that

       ∆U = Φ, in J, ∂U ∂ n = 0, on ∂ J \ i Γ i ∂U ∂ n + T i U = g i , on Γ i (60)
if and only if one satisfies the compatibility condition (in a weak sense)

i Γi

g i = b J Φ (61)
Moreover, W i being defined by (11) and (12), any solution of (60) satisfies

1 c i Γi U - 1 c i+1 Γi+1 U = b J W i Φ + N j=1 Γj W i g j (62)
Proof. The existence-uniqueness proof is a classical exercise about Lax-Milgram lemma and Poincaré-Virtinger's inequality (the important point is that T i :

H 1 2 (Γ i ) → H -1 2 (Γ i
) is a positive symmetric operator whose kernel contains the space of constant functions -see appendix A.1). The compatibility condition (61) is obtained by integrating the first equation of (60), using Green's formula and the symmetry of T i (see again appendix A.1).

To obtain (62), we multiply the equation for U by W i and integrate over J. Using Green's formula twice, and the fact that W i is harmonic, we get

j Γj ∂W i ∂ n U - ∂U ∂ n W i = b J W i Φ
Using the boundary conditions on Γ j for the Neumann traces of U and W i together with the symmetry of T i , we obtain (62).

Lemma 3.3. Given (δ i ) 1 i N -1 ∈ H 1 loc (R * + ) and α ∈ L 2 loc (R * +
) such that δ i (0) = 0 for any i, and given

(f i ) 1 i N -1 ∈ H 1 (]0, +∞[) and (g i ) 1 i N -1 ∈ L 2 (]0, +∞[) , there exists a unique family (u i ) 1 i N ∈ C 0 (R * + , H 1 (]0, +∞[) ∩ C 1 (R * + , L 2 (]0, +∞[)) such that                    ∂ 2 u i ∂t 2 - ∂ 2 u i ∂s 2 i = 0, in R * + ×]0, +∞[ u i (0, •) = f i , ∂u i ∂t (0, •) = g i , N i=1 c i ∂u i ∂s i (•, 0) = α(•) u i+1 (•, 0) -u i (•, 0) = δ i (•), ∀1 i N -1 (63)
Moreover, the norm of the solution is bounded by the norm of the Cauchy data, the norms of the Kirchhoff data and √ t.

Proof. The idea is to say that solution of problem (63) can be decomposed into three functions:

• the solution (u 0 i ) of the following problem:

                     ∂ 2 u 0 i ∂t 2 - ∂ 2 u 0 i ∂s 2 i = 0, in R * + ×]0, ∞[ u 0 i (0, •) = f i , ∂u 0 i ∂t (0, •) = g i , N i=1 c i ∂u 0 i ∂s i (•, 0) = 0 u 0 i+1 (•, 0) -u 0 i (•, 0) = 0, ∀1 i N -1
This problem admits a unique solution

(u 0 i ) 1 i N ∈ C 0 (R * + , H 1 (]0, +∞[)∩ C 1 (R * + , L 2 (]0, +∞[))
(and one has a priori estimates about the solution). • the solution (u 1 i ) of the following problem:

             ∂ 2 u 1 i ∂t 2 - ∂ 2 u 1 i ∂s 2 i = 0, in R * + ×]0, +∞[ u 1 i (0, •) = 0, ∂u 1 i ∂t (0, •) = 0, u 1 i (•, 0) = j<i δ j (•), ∀1 i N
The solution of this system is simply written as (extending each δ j (τ ) to 0 when τ < 0):

u 1 i (t, s i ) = j<i δ j (t -s i ) INRIA and it satisfies u 1 i+1 (•, 0) -u 1 i (•, 0) = δ i (•) Hypothesis on each δ j ensure that (u 1 i ) 1 i N ∈ C 0 (R * + , H 1 (]0, +∞[) ∩ C 1 (R * + , L 2 (]0, +∞[))
. However, we have

N i=1 c i ∂u 1 i ∂s i (•, 0) = - N i=1 j<i c i δ ′ j (•)
• the solution (u 2 i ) of the following problem:

                     ∂ 2 u 2 i ∂t 2 - ∂ 2 u 2 i ∂s 2 i = 0, in R * + ×]0, +∞[ u 2 i (0, •) = 0, ∂u 2 i ∂t (0, •) = 0, N i=1 c i ∂u 2 i ∂s i (•, 0) = α(•) + N i=1 j<i c i δ ′ j (•) u 2 i+1 (•, 0) -u 2 i (•, 0) = 0, ∀1 i N -1
Solution of this equation can be written as follow:

u 2 i (t, s i ) = - 1 c j   t-si 0 α(τ )dτ + N i=1 j<i c i δ j (t -s i )  
Under the hypothesis about α and each δ j , one has

(u 2 i ) 1 i N ∈ C 0 (R * + , H 1 (]0, +∞[)∩ C 1 (R * + , L 2 (]0, +∞[)).
Finally, one can check that u i = u 0 i + u 1 i + u 2 i satisfies problem (63), and uniqueness is obtained thanks to uniqueness of the problem with classical Kirchoff conditions (if there exists two solutions of (63), the difference satisfies the wave equation with classical Kirchhoff conditions and null initial conditions). Now, we compute for a given time t ∈ R * + the differents norms of the solution of problem (63). One can easily check that

u i (t, •) L 2 (]0,+∞[) u 0 i (t, •) L 2 (]0,+∞[) + u 1 i (t, •) L 2 (]0,+∞[) + u 2 i (t, •) L 2 (]0,+∞[) ∂u i ∂t (t, •) L 2 (]0,+∞[) ∂u 0 i ∂t (t, •) L 2 (]0,+∞[) + ∂u 1 i ∂t (t, •) L 2 (]0,+∞[) + ∂u 2 i ∂t (t, •) L 2 (]0,+∞[) ∂u i ∂s i (t, •) L 2 (]0,+∞[) ∂u 0 i ∂s i (t, •) L 2 (]0,+∞[) + ∂u 1 i ∂s i (t, •) L 2 (]0,+∞[) + ∂u 2 i ∂s i (t, •) L 2 (]0,+∞[)
By using some classical estimates, we can say that the norm u 0 i (t, •) L 2 (]0,+∞[) is bounded by square root of order 1 polynom (in t) depending on the Cauchy datas, and the norm of its derivatives is bounded by a constant depending only on Cauchy data. For the functions u 1 i and u 2 i , by using explicit computation, one has:

• for terms u 1 i :

u 1 i (t, •) L 2 (]0,+∞[) j<i c i δ j L 2 (]0,t[) ∂u 1 i ∂t (t, •) L 2 (]0,+∞[) j<i c i δ ′ j L 2 (]0,t[) ∂u 1 i ∂s i (t, •) L 2 (]0,+∞[) j<i c i δ ′ j L 2 (]0,t[)
• for terms u 2 i :

u 2 i (t, •) L 2 (]0,+∞[) 1 c j   √ t α L 2 (]0,t[) + N i=1 j<i c i δ j L 2 (]0,t[)   ∂u 2 i ∂t (t, •) L 2 (]0,+∞[) 1 c j   α L 2 (]0,t[) + N i=1 j<i c i δ ′ j L 2 (]0,t[)   ∂u 2 i ∂s i (t, •) L 2 (]0,+∞[) 1 c j   α L 2 (]0,t[) + N i=1 j<i c i δ ′ j L 2 (]0,t[)  
and the proof is complete.

Remark 3.1. In the case where α ∈ L 2 (]0, +∞[) and δ i ∈ H 1 (][0, +∞), we can see that the L 2 norm of derivatives of solution is uniformly bounded over time.

Existence and uniqueness

According to ((33), (34)) b J and (59), we can apply, for a given

t ∈ R * + , lemma 3.2 with U = U k (t, •), Φ = ∂ 2 U k-2 ∂t 2 (t, •) and g i = g k-1 i (t, •) with g k-1 i (t, •) = ∂u k-1 i ∂s i (t, 0) + k/2 l=1 ∂ 2l+1 u k-2l-1 i ∂t 2l ∂s i (t, 0) σ 2l i (2l)! + k/2 l=1 ∂ 2l u k-2l i ∂t 2l (t, 0) σ 2l-1 i (2l -1)! + k/2 l=1 p 1 ∂ 2l D i,p (U k-2l (t, •)) ∂t 2l e " - pπσ i c i " ∂d 2l i,p ∂s i (σ i ) w p,i (64)
where the index k -1 in g k-1 i is "justified" by the fact that g k-1 i is known explicitly when the u k-1 i and the U m for m k -1 are known. Writing (61)

INRIA

gives, once we made the change of index k = k + 1,

N i=1 c i ∂u k i ∂s i (t, 0) = b J ∂ 2 U k-1 ∂t 2 - N i=1 c i (k+1)/2 l=1 ∂ 2l+1 u k-2l i ∂t 2l ∂s i (t, 0) σ 2l i (2l)! - N i=1 c i (k+1)/2 l=1 ∂ 2l u k+1-2l i ∂t 2l (t, 0) σ 2l-1 i (2l -1)! (65) Remark 3.2. One could see that if we take (σ ′ i ) 1 i N such that σ ′ i σ i
, and if we define J ′ as

J ′ = J ∪ N i=1 x ∈ R 2 / /σ i x • t i < σ ′ i and |x • n i | < c i 2 , one should have N i=1 c i ∂u k i ∂s i (t, 0) = b J ∂ 2 U k-1 ∂t 2 - N i=1 c i (k+1)/2 l=1 ∂ 2l+1 u k-2l i ∂t 2l ∂s i (t, 0) σ 2l i (2l)! - N i=1 c i (k+1)/2 l=1 ∂ 2l u k+1-2l i ∂t 2l (t, 0) σ 2l-1 i (2l -1)!
This point can be proved by using the modal expansion of U k-1 on each rectangle

( s i , ν i ) ∈]σ i , σ ′ i [×] -c i /2, c i /2[. Given 1 i
N , integrating U k on Γ i by using modal expansion (56) and "Dirichlet" condition (55-i) -"Neumann" condition (55-ii) gives

1 c i Γi U k (t, •) = u k i (t, 0) + ∂u k-1 i ∂s i (t, 0)σ i + k/2 l=1 ∂ 2l+1 u k-2l-1 i ∂t 2l ∂s i (t, 0) σ 2l+1 i (2l + 1)! + k/2 l=1 ∂ 2l u k-2l i ∂t 2l (t, 0) σ 2l i (2l)! (66)
Then using (62

) with U = U k , Φ = ∂ 2 U k-2
∂t 2 and g k-1 i defined in (64), one has, for 1 i N -1

u k i (t, 0) -u k i+1 (t, 0) = b J ∂ 2 U k-2 ∂t 2 W i + n j=1 Γj W i g k-1 j + ∂u k-1 i+1 ∂s i+1 (t, 0)σ i+1 - ∂u k-1 i ∂s i (t, 0)σ i + k/2 l=1 ∂ 2l+1 u k-2l-1 i+1 ∂t 2l ∂s i+1 (t, 0) - k/2 l=1 ∂ 2l+1 u k-2l-1 i ∂t 2l ∂s i (t, 0) + k/2 l=1 ∂ 2l u k-2l i+1 ∂t 2l (t, 0) σ 2l i+1 (2l)! - k/2 l=1 ∂ 2l u k-2l i ∂t 2l (t, 0) σ 2l i (2l)! (67)
For each k, we have succeeded to decouple the calculation of u k i since jump conditions (67) and average condition (65) written for k = k + 1 are sufficient, when associated to equations (63), to determine u k i uniquely (lemma 3.3). As the solution of problem (60

) with Φ = ∂ 2 U k-2 ∂t 2 and g i = g k-1 i , U k is defined
up to an additive constant. To fix this constant we can use again (55-i) (in a symmetric way with respect to i):

N i=1 1 c i Γi U k (t, •) = N i=1 u k i (t, 0) + N i=1 ∂u k-1 i ∂s i (t, 0)σ i + N i=1 k/2 l=1 ∂ 2l+1 u k-2l-1 i ∂t 2l ∂s i (t, 0) σ 2l+1 i (2l + 1)! + N i=1 k/2 l=1 ∂ 2l u k-2l i ∂t 2l (t, 0) σ 2l i (2l)! (68)
Finally, we obtain a problem equivalent to ((31), (32)), ((33), (34)) b J , (55-i) and (59) by replacing (55-i) by ( 65), ( 67) and (68). More precisely Theorem 3.4. The following two propositions are equivalent (for the clarity of notation, we omit to mention again the functional setting): 31), (32)), ((33), (34)) b J , (55-i) and (59) 31), (32), ( 65), (67)) and ((33), (34)) b J , (59), (68) with g k-1 i defined as (64)

(i) (u k 1 , . . . , u k N , U k ), k 0 is solution of ((
(ii) (u k 1 , . . . , u k N , U k ), k 0 is solution of ((
Proof. We just proved the implication (i) ⇒ (ii). We will prove the implication (ii) ⇒ (i). Let (u k 1 , . . . , u k N , U k ), k 0 be a solution of ((31), ( 32), ( 65), (67)) and ((33), (34)) b J , (59), (68) with g k-1 i defined as (64). The only point we have to prove is (55-i) is satisfied. In fact, by taking the modal extension of U k

INRIA

on Γ i , we can rely D i,0 (U k (t, •)) and the integration of U k over Γ i . However, (68) gives some information about the sum of the integrals of U k over the Γ i 's, and (62) gives some information about the difference of the integrals of U k over two consecutive Γ i 's. By taking some linear combination, we can retrieve the integral of U k over a given Γ i , and by identification, (55-i) is satisfied.

Next, we show that ((31), ( 32), ( 65), (67)) and ((33), (34)) b J , (59), ( 68), with g k-1 i defined as (64), admits a unique solution (u k 1 , . . . , u k N , U k ), k 0 , by induction on k ∈ N. The case k = 0. With the convention of 2.2, we see from ((31), ( 32), ( 65), (67)) that u 0 i is, as expected, the solution of the 1D wave equation with classical Kirchhoff conditions at the node (see ( 6), ( 7)).

Moreover, we see from ((33), (34)) b J , (59) that U 0 solves (60) with Φ = 0 and g i = 0, which implies that U 0 (t, •) is constant. Next, (68) gives

U 0 (t, •) = 1 N N i=1 1 c i Γi U 0 (t, •) = 1 N N i=1 u 0 i (t, 0) on J
The general case k 1. Assume that (u l 1 , . . . , u l N , U l ), l k -1 are known, then, according to theorem 3.4,

• We compute g k-1 i thanks to (64),

• We determine (u k i ) 1 i N as the unique solution of the 1D transmission problem (63) with the transmission conditions (67, 68) (cf. lemma 3.3),

• We determine U k as the solution, cf. lemma 3.2, of the boundary value problem ((33, 34) b J , 59, 68). One must of course check the compatibility relation (61) (written for

g i = g k-1 i and Φ = ∂ 2 U k-2
∂t 2 ), which is a consequence of (65).

Finally, regrouping the above results with theorem 3.1 and 3.4, we have proved the following theorem Theorem 3.5. There exists a unique family

(u k 1 , . . . , u k n , U k ) ∈ C 1,2 (]0, L 1 [) × • • • × C 1,2 (]0, L N [) × C 1,2,loc ( J), k 0
satisfying (31,32,33,34), the matching conditions ( 55) and the growth condition (54).

Remark 3.3. In fact, by construction, one can check that, for a given k 1, the datas α and δ i of problem (63) depends on

∂ 2 u k-1 ∂t 2
and ∂u k-1 ∂s . By recurrence, one can check that if we want to buid u k , the real regularity on u 0 must be

u 0 i ∈ C 0 (R * + , H k+1 (]0, +∞[) ∩ C 1 (R * + , H k (]0, +∞[))
and since the problem satisfied by u 0 is a classical problem, one can check that we should have

f ε i ∈ H k+1 (]0, +∞[) g ε i ∈ H k (]0, +∞[)

Error estimates

In this section, we will give some error estimates between the solution of exact problem (1) and a function built from our matched asymptotic expansions (27, 28). We recall here the natural energy (4) associated to the exact problem:

E ε (t, u ε ) = 1 2ε Ω ε |∇u ε (t, x)| 2 + ∂u ε ∂t (t, x) 2 dx
whose derivative is given by (denoting

:= ∂ 2 ∂t 2 -∆): ∂E ε (t, u ε ) ∂t = 1 ε Ω ε u ε (t, x) ∂u ε ∂t (t, x)dx + 1 ε ∂Ω ε ∂u ε ∂ n (t, σ) ∂u ε ∂t (t, σ)dσ = 0 Then we have E ε (t, u ε ) = E ε (0, u ε ).
Under the hypothesis 1.2, one can see that, for ε small enough,

E ε (0, u ε ) = N i=1 c i 2 +∞ 0 ∂f ε ∂s i (t, s i ) 2 + |g ε (t, s i )| 2 ds i
This last relation explain the 1 ε multiplicative coefficient in the definition of the energy. To be able to give error estimates, let us introduce a C ∞ cut-off function χ defined on R satisfying the three following points:

• χ : R → [0, 1] • χ(] -∞, 1]) = 0 • χ([2, +∞[) = 1
From this 1D cut-off function, we can define N bi-dimensional cut-off functions

(χ ε i ) 1 ß N defined as • χ ε i (x) = χ(ϕ(ε) -1 x • t i ) when x • t i > 0 • χ ε i (x) = 0 elsewhere Since ϕ(ε)ε -1
is continuous and tends to ∞ when ε tends to zero, one can assume that, for ε < ε 0 , ε -1 ϕ(ε) > max σ i . This ensures that χ ε i χ ε j = 0, for ε < ε 0 and i = j. Once we introduced our cut-off functions, we can define our approximated solution by

ũε n (t, x) = N i=1 χ ε i (x) n k=0 ε k u k i (t, x • t i ) + 1 - N i=0 χ ε i (x) n k=o ε k U k (t, ε -1 x) (69)
One can easily see that, for ε small enough, and by using some Taylor expansion, we get the following inequality for the intial time estimate:

E ε (0, u ε -ũε n ) = ε n+1 f ′ 2 L 2 (]0,+∞[) + g
We are first looking for the normal derivative of the approximated function on boundary ∂Ω ε : because of the properties of the cut-off funtions, we have

∂ ũε n ∂ n (t, x) = 0 ∀x ∈ ∂Ω ε then it gives ∂(ũ ε n -u ε ) ∂ n (t, x) = 0 ∀x ∈ ∂Ω ε
On can see, thanks to the Green-Riemann formula, that the derivate of the energy E ε (•, u εũε n ) with respect to time gives

∂E ε (•, u ε -ũε n ) ∂t (t) = 1 ε Ω ε (u ε -ũε n )(t, x) ∂(u ε -ũε n ) ∂t (t, x)dx (71)
Next point is to compute the term ũε n (t, x), one has

(ũ ε n -u ε )(t, x) = ũε n (t, x) = N i=1 χ ε i (x) n k=0 ε k u k i (t, x) + 1 - N i=1 χ ε i (x) n k=0 ε k U k (•, ε -1 •) (t, ε -1 x) + 2 N i=1 ϕ(ε) -1 χ ′ ϕ(ε) -1 x • t i n k=0 ε k ∂u k i ∂s i (t, x) -ε k-1 ∂U k ∂ s i (t, ε -1 x) + N i=1 ϕ(ε) -2 χ" ϕ(ε) -1 x • t i n k=0 ε k u k i (t, x) -ε k U k (t, ε -1 x) (72)
On the relation (72), we treat separately the four lines:

• The first line, that is the simpliest one. For each function u k i , proposition 2.1 ensures that u k i = 0, the the first line vanishes. • The second line. Each function

U k does not satisfy U k (•, ε -1 •) = 0.
However, thanks to (33), one has

U k (•, ε -1 •) = ∂ 2 U k ∂t 2 - 1 ε 2 ∂ 2 U k-2 ∂t 2
then the second line becomes

1 - N i=1 χ ε i (x) ε k ∂ 2 U k ∂t 2 (t, ε -1 x) + ε k-1 ∂ 2 U k-1 ∂t 2 (t, ε -1 x) (73)
• The two groups of last lines (lines 3-4 and lines 5-6), that we can deal with the same manner. The main point is to use the relations (53, 55) associated to corollary 2.4, and the fact that χ ′ (ϕ(ε) -1 •) and χ"(ϕ(ε) -1 •) do vanish when x • t i ∈]ϕ(ε), 2ϕ(ε)[. We do the computations with the fourth line (treatment of this line is simplier than treatment of the third line). For U k , one starts with the modal expansion of corollary 2.4, and let us call Π k the associated function given by the sum of terms for p 1. Since χ"(ϕ(ε) -1 s i ) do vanish for s n ϕ(ε), Π k is a finite sum of exponentially decreasing functions and one can see that, given q ∈ N, that we can bound uniformly Π k for 0 k n by a constant (depending of the choice of φ, q and n) times ε q . For the function u k i , since χ"(ϕ(ε) -1 s i ) do vanish for s i 2ϕ(ε), which tends to 0, we simply use a Taylor expansion on s n = 0 with Lagrange remainder at order nk (the reason is that the l-th derivative coefficient of the expansion appears, thanks to the relations (55), in the modal expansion of the function U k+l ): one gets

u k i (t, s i ) = n-k l=0 s l i l! ∂ l u k i ∂s l i (t, 0) + si 0 1 (n -k)! ∂ n+1-k u k i ∂s n+1-k i (t, σ)(s i -σ) n-k dσ (74)
and by using, as expected, the relations (55), the fifth and sixth lines become

n i=1 ϕ(ε) -2 χ" ϕ(ε) -1 x • t i n k=0 ε k I i,n,k + o(ε q ) (75)
with

I i,n,k = x•ti 0 1 (n -k)! ∂ n+1-k u k i ∂s n+1-k i (t, σ)(x • t i -σ) n-k dσ and, for x • t i 2ϕ(ε): |I i,n,k | C i,n,k ϕ(ε) n-k+1
with C i,n,k which does not depend on ε. In a same way, the third and fourth lines become

n i=1 ϕ(ε) -1 χ ′ ϕ(ε) -1 x • t i ε k ∂u k i ∂s i (t, x • t i ) + n-1 k=0 ε k x•ti 0 1 (n -k -1)! ∂ n-k+1 u k i ∂s n-k+1 i (t, σ)(x • t i -σ) n-k dσ (76)
with the same type of increase. Remark 3.4. One can easily see that the constant use for bound Π k blows up when φ(ε) tends to ε (in sense of functions) . This is due to the fact that the exponentials exp(-pπs i /ε) are not small when s i tends to ε.

Next, we multiply ũε n (t, x) by the derivate of ũε n with respect to time, and we integrate over the space domain Ω ε , after dividing by ε to get the derivative of the energy (see back (71)). The main point is to use the fact that, for any function Ψ ∈ L 2 (Ω ε ) supported for x • t i 2ϕ(ε) (whose proof is done by INRIA using the Cauchy-Schwartz inequality and doing some calculs similar of those of lemma 3.7 of [START_REF] Joly | Propagation of an acoustic wave in a junction of two thin slots[END_REF]):

Ω ε Ψ(x) ∂ ũε n ∂t (t, x) Ψ L 2 (Ω ε ) ∂ ũε n ∂t (t, •) L 2 (Ω ε )∩|x•ti 2ϕ(ε)| C Ψ L 2 (Ω ε ) E ε (t, ũε n ) (77)
We apply (77) with Ψ given by various terms of ( 73), ( 75) and ( 76), and after some tedious computations, one gets that Lemma 3.6. There exist a function M depending on time, n and Cauchy data (f, g) such that, for ε small enough,

∂E ε (•, ũε n -u ε ) ∂t (t) φ(ε) n-1 2 M (t) E ε (t, ũε n -u ε ) (78)
By integrating the relation (78), and by using error estimate about initial state (70), one has

E ε (t, ũε n -u ε ) E ε (0, ũε n -u ε ) + φ(ε) n-1 2 t 0 M (τ ) E ε (τ, ũε n )dτ (79)
To conclude, one has to use the following variant of the Gronwall lemma: Lemma 3.7 (admitted -see [START_REF] Bihari | A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations[END_REF] for a complete proof in a more general case).

Let C > 0 given, and φ(t), m(t) be two continuous positive functions defined on [0, T ], satisfying

∀t ∈]0, T [, φ(t) C + t 0 m(τ ) φ(τ )dτ then we have ∀t ∈]0, T [, φ(t) √ C + 1 2 t 0 m(τ )dτ 2
Using this lemma for and (79) leads to the following formula

E ε (t, ũε n -u ε ) E ε (0, ũε n -u ε ) + 1 2 φ(ε) n-1 2 t 0 M (τ )dτ 2 (80)
Taking ( 80) with looking at the fact that

E ε (0, ũε n -u ε ) Cε n+1 φ(ε) n-1 2 leads to E ε (t, ũε n -u ε ) C(f, g, t)φ(ε) 2n-1 (81)
Remark 3.5. This result is underoptimal. In fact, we can see that the term responsible for the behaviour φ(ε) 2n-1 is "located" in the junction zone. The idea is to get some error estimates on the slots zones, far from the junction zone. This is the object of the following corollary Corollary 3.8. Given n ∈ N. Let us assume that the terms of the asymptotic expansion may be built up to term n + 2 (this implies that

f ε i ∈ H n+3 (]0, +∞[) and g ε i ∈ H n+2 (]0, +∞[)). Given (δ i > 0) 1 i N ,
Ω ε i,δ = {x ∈ Ω ε i such that x • t i δ i }
• the following 2D function

u ε i,n (t, x) = N n=0 ε n u n i (t, x • t i ) for t ∈ R * + and x ∈ Ω ε i,δ
Then, one has the following error estimate

N i=1 1 ε Ω ε i,δ ∂(u ε -u ε i,n ) ∂t 2 + ∇(u ε -u ε i,n ) 2 C(f, g, t)ε 2n+2
Proof. By using inequality (ab) 2 2a 2 + 2b 2 , one has

N i=1 1 ε Ω ε i,δ ∂(u ε -u ε i,n ) ∂t 2 + ∇(u ε -u ε i,n ) 2 N i=1 2 ε Ω ε i,δ ∂(u ε -u ε i,n+2 ) ∂t 2 + ∇(u ε -u ε i,n+2 ) 2 + N i=1 2 ε Ω ε i,δ ∂(u ε i,n+2 -u ε i,n ) ∂t 2 + ∇(u ε i,n+2 -u ε i,n ) 2 (82)
To treat the first term of the right member of (82), we see that the sums of the differents integrals is bounded by the integral on the whole domain Ω ε , and this integral is no other than the energy (the fact that δ i is fixed let us allow to choose ε small enough such that 2φ(ε) δ i ∀i, i.e. the function ũε n coincides with u ε i,n+2

on each domain Ω ε i,δ ). We use then (81) with n+2 instead of n. The second term is easier to treat: one can easily see that

u ε i,n+2 -u ε i,n = ε n+1 u n+1 i + ε n+2 u n+2 i with u n+1 i
and u n+2 i known: one can bound the energy of ε n+1 u n+1 i + ε n+2 u n+2 i by C ′ (f, g, t)ε 2n+2 , for ε small enough. Finally, one has

N i=1 1 ε Ω ε i,δ ∂(u ε -u ε i,n ) ∂t 2 + ∇(u ε -u ε i,n ) 2 2C(f, g, t)φ(ε) 2n+3 + 2C ′ (f, g, t)ε 2n+2 (83)
Now, we choose φ : ε → ε 2n+2 2n+3 . One gets that φ(ε) 2n+3 = ε 2n+2 and the proof is complete. Remark 3.6. In this proof, one can check why we need to get two orders further (one order would be not enough to get optimal estimate). In this section, once we proved by the theorem 3.5 that the functions u k i and U k are uniquely defined, we explain how we can get the improved 1D problem (18, 19, 70). In the following, we shall denote by ũε i the function equal to u 0 i + εu 1 i on the i th slot. We can see easily that

                   ∂ 2 ũε i ∂t 2 - ∂ 2 ũε i ∂s 2 i = 0 in R * + ×]εσ i , +∞[ ũε i = f ε + O(ε 2 ) on {0}×]εσ i , +∞[ ∂ ũε i ∂t = g ε + O(ε 2 ) on {0}×]εσ i , +∞[ (84)
One can see that, neglecting the O(ε 2 ) term and for ε small enough, that ũε i satisfies the same Cauchy data as u ε . Next point to conclude is to link the values of (ũ ε i ) 1 i N near s i = 0. Remark 4.1. Ideally, the link of the values of (ũ ε i ) 1 i N would be at s i = 0. However, one can see that associated problem would be ill-posed in terms of energy (for example, with a negative term of the form 1/ε).

Average condition

Writing (65), for k = 0 and k = 1, gives respectively:

N i=1 c i ∂u 0 i ∂s i (t, 0) = 0 (85-i) N i=1 c i ∂u 1 i ∂s i (t, 0) = b J ∂ 2 U 0 ∂t 2 (t, x)dx - N i=1 σ i c i ∂ 2 u 0 i ∂t 2 (t, 0) (85-ii)
In the proof of theorem 3.4, we show that U 0 is constant and equal to N -1 u 0 i . Rewritting (85-ii) leads to

N i=1 c i ∂u 1 i ∂s i (t, 0) + N i=1 σ i c i ∂ 2 u 0 i ∂s 2 i (t, 0) = | J| N N i=1 ∂ 2 u 0 i ∂t 2 (86)
Finally, one gets

N i=1 c i ∂ ũε i ∂s i (t, εσ i ) = N i=1 c i ∂u 0 i ∂s i (t, 0) + ε N i=1 c i σ i ∂ 2 u 0 i ∂s 2 i (t, 0) + ε N i=1 c i ∂u 1 i ∂s i (t, 0) + O(ε 2 ) = ε b J ∂ 2 U 0 ∂t 2 (t, x)dx + O(ε 2 ) = ε | J| N N i=1 ∂ 2 u 0 i ∂t 2 (t, 0) + O(ε 2 ) = ε | J| N N i=1 ∂ 2 ũε i ∂t 2 (t, εσ i ) + O(ε 2 )
Let us now use notations (14-i) and (14-ii) of section 1.3.3, and let us introduce the vector 1 of R N equal to (1, . . . , 1) T . Let us also introduce the canonical scalar product in

R N ∀A, B ∈ (R N ) 2 , (A, B) = N i=1 A i B i
The last line of the previous equation can be rewritten, neglecting the

O(ε 2 ) term C∂ S Ũ ε (t), 1 (1, 1) 1 = εA ∂ 2 ∂t 2 Ũ ε (t) (87)
where C and A are the matrices introduced in the section 1.3.3.

Jump conditions

We now take the problem (60) satisfied by U 1 (t, •) with Φ = 0 and g i = ∂u 0 i ∂si , and difference of (66) for k = 0, 1 and i = j, j + 1 gives u 0 j (t, 0)u 0 j-1 (t, 0) = 0 (88-i)

u 1 j (t, 0) + σ j ∂u 0 j ∂s j (t, 0) -u 0 j+1 (t, 0) -σ j+1 ∂u 0 j+1 ∂s j+1 = 1 c j Γj+1 U 1 (t, •) - 1 c j+1 Γj+1 U 1 (t, •) (88-ii)
The idea now is to express right member side of (88-ii) with respect to u 0 . One can see first that this member does not depend on the additive constant added to U 1 . Then, (85-i) gives that, since c 1

∂u 0 1 ∂s1 (t, 0), . . . , c N ∂u 0 N ∂sN (t, 0) • 1 = 0, c 1 ∂u 0 1 ∂s 1 (t, 0), . . . , c N ∂u 0 N ∂s N (t, 0) T ∈ Im P T (89)
So there exists a vector Φ 0 (t) = (Φ 0 1 (t), . . . ,

Φ 0 N -1 (t)) ∈ R n-1 such that c 1 ∂u 0 1 ∂s 1 (t, 0), . . . , c N ∂u 0 N ∂s N (t, 0) T = P T Φ 0 (t)

INRIA

By the definition of P T , one can see, modulo an additive constant, that

U 1 (t, •) = N -1 i=1 Φ 0 i (t)W i
with W i defined in (11), then one has

1 c j Γj+1 U 1 (t, •) - 1 c j+1 Γj+1 U 1 (t, •) = N -1 i=1 K i,j Φ 0 i (t) (90)
with K i,j defined in (13). Combining (88-i) + ε(88-ii) with (90) leads to (using proposition 1.4 that ensures K is invertible):

P Ũ ε + O(ε 2 ) = εKΦ 0 =⇒ εΦ 0 = K -1 P Ũ ε + O(ε 2 ) (91)
Looking for (89) make us with to say that C∂ S Ũ ε + O(ε 2 ) • 1, and this point is not certain. One better idea is to say that

  C∂ S Ũ ε (t) - C∂ S Ũ ε (t), 1 (1, 1) 1 
 • 1 = 0 (92)
Then we have, since

C∂ S Ũ ε - (C∂S Ũ ε (t),1)
(1,1)

1 = c 1
C∂ S Ũ ε - C∂ S Ũ ε (t), 1 (1, 1) 1 = P T Φ 0 + O(ε)
combining with (91) and neglecting the O(ε 2 ) term leads to

ε    C∂ S Ũ ε - C∂ S Ũ ε (t), 1 (1, 1) 1 
  = P T K -1 P Ũ ε (93)
Finally, dividing (93) by ε and introducing J = P T K -1 P leads to

C∂ S Ũ ε - C∂ S Ũ ε (t), 1 (1, 1) 1 = 1 ε J Ũ ε (94)

Conclusion

Finally, adding (94) and (87) leads to the Improved Kirchhoff conditions (19):

C∂ S Ũ ε = 1 ε J Ũ ε + εA ∂ 2 ∂t 2 Ũ ε (t) (95)

Analysis of Improved Kirchhoff conditions

Here, we have to show that problem (84) with Improved Kirchhoff conditions (19) gives a well-posed problem whose solution differs from the solution of the exact problem (1) by a smaller error than the error between the limit solution and the exact solution. In this way, we show that the the solution of this improved model admits an asymptotic expansion, whose two first terms (and not only the first one) are equal to the two first terms of the matched asymptotic expansion of the slots terms for the solution of the exact problem.

Asymptotic expansions

One starts from the following ansatz:

Ansatz 4.2. For 1 i N , there exists a family of functions

(ũ k i ) k∈N defined on R + ×]0, +∞[ such that ũε i (t, s i ) = k∈N ε k ũk i (t, s i ) on R + ×]εσ i , +∞[ (96)
The idea is to prove that the error between the solution of the Improved model and its truncated approximation is small, and the two first terms ũ0 i , ũ1 i are the same than the two first terms u 0 i , u 1 i of the development of exact solution.

Equations This is the simpliest part. Using ansatz (96) in system (84), separating in powers of ε leads to the fact that ũk i satisfy the equations (31-ii), (32-i) and (32-ii) on each domain ]εσ i , +∞[. Tending ε to 0 ensure us that ũk i satisfy the equations (31-ii), (32-i) and (32-ii) on the same domain as the functions u k i .

Matching conditions One start from the Improved Kirchhoff conditions (19), and do the scalar product with 1. Then one gets that

N i=1 c i ∂ ũε i ∂s i (t, εσ i ) = ε | J| N N i=1 ∂ 2 ũε i ∂t 2 (t, εσ i ) (97)
Using ansatz (96) and taking Taylor expansion of each function ũn i with respect to s i at the point s i = 0 leads to

N i=1 c i n∈N ε n k∈N ε k σ k i k! ∂ k+1 ũn i ∂s k+1 i (t, 0) = ε | J| N N i=1 n∈N ε n k∈N ε k σ k i k! ∂ k+2 ũn i ∂s k i t 2 (t, 0)
One can see that we have infinite polynomial sum with respect to ε. Taking the terms in ε 0 and ε 1 leads to this two relations

N i=1 c i ∂ ũ0 i ∂s i (t, 0) = 0 N i=1 c i ∂ ũ1 i ∂s i (t, 0) + N i=1 σ i c i ∂ 2 ũ0 i ∂s 2 i (t, 0) = | J| N N i=1 ∂ 2 ũ0 i ∂t 2 (t, 0)
which is nothing but (85-i) and (86). Dealing with Dirichlet jumps is more technical, however, using some remarks about the Neumann jump will help us. Using again the remark that 1 ∈ Ker(J ), 97 (under its matricial form) multiplied by 1 gives

1 T C∂ S Ũ ε (t) 1 = ε 1 T A ∂ 2 Ũ ε ∂t 2 (t) 1
One better idea is to say that

C∂ S Ũ ε (t), 1 (1, 1) 1 = ε A ∂ 2 Ũ ε ∂t 2 (t), 1 (1, 1) 1 (98)
INRIA and to say the fact that scalar product of 1 and any vector which we removed its average part is equal to zero. Then, is exists two functions Φ ε and Ψ ε depending on time (and ε) such that

C∂ S Ũ ε (t) - C∂ S Ũ ε (t), 1 (1, 1) 1 = P T Φ ε (t) (99-i) A ∂ 2 Ũ ε ∂t 2 (t) - A ∂ 2 Ũ ε ∂t 2 (t), 1 (1, 1) 1 = P T Ψ ε (t) (99-ii)
One can even see that Φ 0 (t) is nothing but the one defined in section 4.1.2 by replacing each u 0 i by ũ0 i . Now, taking ε( 19) -(98), using (99-i) and (99-ii), leads to

εP T Φ ε (t) = J Ũ ε (t) + ε 2 P T Ψ ε (t) (100)
Now, using the fact that P T is injective and the definition of J , and using the fact that K is invertible leads to:

εKΦ ε (t) = P Ũ ε (t) + ε 2 KΨ ε (t) (101)
Finally, using ansatz (96) and developments of Φ ε and Ψ ε with respect to ε, using Taylor expansion of each function ũn i with respect to s i at the point s i = 0 and looking at the terms in ε 0 and ε 1 leads to (88), writing ũ0 i and ũ1 i instead of u 0 i and u 1 i and using (90) for the right member side of (88-ii).

Conclusion

One can see first that (ũ 0 i ) 1 i N and (u 0 i ) 1 i N share the same equation, the same Cauchy data and the same jump conditions. Lemma 3.3 ensures that the two families are the same one (this point is normal, since the limit problem does only depend on the topology of the graph). Then, one can see that (ũ 1 i ) 1 i N and (u 1 i ) 1 i N share the same equation, the same Cauchy data and the same jump conditions. Lemma 3.3 ensures again that the two families are the same one (this point is the expected bonus point).

Matching conditions under the general form Let us keep ansatz (96), and for a given k in N, we denote by ∂ s l Ũ k (t) the vector in R N whose i-th coordinate is equal to ∂ l ũk i ∂s l i (t, 0). Injecting this ansatz in (95) and taking a Taylor development of each vector Ũ k,ε with respect to each s i gives, denoting S the diagonal matrix whose i-th coefficient is equal to σ i :

1 ε J n∈N ε n k∈N ε k k! S k ∂ s k Ũ n (t) +εA n∈N ε n k∈N ε k k! S k ∂ t 2 s k Ũ n (t) = C n∈N ε n k∈N ε k k! S k ∂ s k+1 Ũ n (t) (102)
Identifying coefficients of (102) with the same power of ε gives:

• power -1 of ε gives J Ũ 0 (t) = 0

• power 0 of ε gives

J S∂ s Ũ 0 (t) + Ũ 1 (t) = C∂ s Ũ 0 (t) • power n + 1 of ε, with n ∈ N, gives J n+2 k=0 1 (n + 2 -k)! S n+2-k ∂ s n+2-k Ũ k (t) +A n k=0 1 (n -k)! S n-k ∂ s n-k ∂ 2 Ũ k ∂t 2 (t) = C n+1 k=0 1 (n + 1 -k)! S n+1-k ∂ s n+2-k Ũ k (t)

Error estimates

As for section 3.2, we will give some error estimates between the solution of Improved problem (18,19) and its development. Let us recall the 1D energy (20):

E ε (t, v) = N i=1 c i 2 +∞ εσi ∂v ∂s i (t, s ′ ) 2 + ∂v ∂t (t, s ′ ) 2 ds ′ + 1 2ε (J V ε (t)) • V ε (t) + ε 2 A ∂V ε ∂t (t) • ∂V ε ∂t (t)
How do we get this energy? Let us consider the first line of (18). Let us multiply by a sequence of functions (v i ) 1 i N , and integer over s i ∈]εσ i , +∞[:

0 = N i=1 c i +∞ εσi ∂ 2 ũε ∂t 2 (t, s ′ )v(s ′ ) - ∂ 2 ũε ∂s 2 i (t, s ′ )v(s ′ ) ds ′
On the second part of each integral, we use the Green formula, and one gets

0 = N i=1 c i +∞ εσi ∂ 2 ũε ∂t 2 (t, s ′ )v(s ′ ) + ∂ ũε ∂s i (t, s ′ ) ∂v ∂s i (s ′ ) ds ′ + C∂ S Ũ ε (t) • V ε Finally, using (19) leads to 0 = N i=1 c i +∞ εσi ∂ 2 ũε ∂t 2 (t, s ′ )v(s ′ ) + ∂ ũε ∂s i (t, s ′ ) ∂v ∂s i (s ′ ) ds ′ + 1 ε J + εA ∂ 2 ∂t 2 Ũ ε (t) • V ε (103)
Finally, taking v = ∂ ũε ∂t leads to the derivative of the 1D energy (20). Moreover, one can see that we have, since matrices J and A are symmetric:

∂E ε (t, v) ∂t = N i=1 c i +∞ εσi ∂ 2 v ∂t∂s i (t, s ′ ) ∂v ∂s i (t, s ′ ) + ∂ 2 v ∂t 2 (t, s ′ ) ∂v ∂t (t, s ′ ) ds ′ + 1 ε J V ε (t) + εA ∂ 2 V ε ∂t 2 (t) ∂V ε ∂t (t)

INRIA

Using the Green-Riemann formula, one gets that

∂E ε (t, v) ∂t = N i=1 c i +∞ εσi - ∂v ∂t (t, s ′ ) ∂ 2 v ∂s 2 i (t, s ′ ) + ∂ 2 v ∂t 2 (t, s ′ ) ∂v ∂t (t, s ′ ) ds ′ + 1 ε J V ε (t) + εA ∂ 2 V ε ∂t 2 (t) -C∂ S V ε (t) ∂V ε ∂t (t) (104
) Now suppose that the terms of ansatz (96) exist up to k = n + 1 (this implies that f ∈ H n+2 and g ∈ H n+1 ), and let us build ũε,n the function defined by

ũε,n i (t, s i ) = n k=0 ε k ũk i (t, s i ) on R + ×]εσ i , +∞[
Since each term of the development satisfy time-domain wave equation on each R + ×]εσ i , +∞[, the truncated expansion satisfies also this equation. However, for the node condition, one can see that, when developping (102) with powers of ε, the term in ε n uses Ũ n+1 . More precisely, there exists a vector Φ ε,n (t) bounded with respect of ε, for ε small enough, such that

1 ε J Ũ ε,n (t) + εA ∂ 2 Ũ ε,n ∂t 2 (t) -C∂ S Ũ ε,n = -ε n J Ũ n+1 (t) -ε n+1 Φ ε,n (t) (105)
Using ( 105) in (104) gives

∂E ε (t, ũε -ũε,n ) ∂t (t) = ε n J Ũ n+1 (t) • ∂( Ũ ε -Ũ ε,n ) ∂t (t) + ε n+1 Φ ε,n (t) • ∂( Ũ ε -Ũ ε,n ) ∂t (t) (106)
Finally, using that some properties on matrices J and A let us allow to express the following result: there exist two vectors Φ ε,n J and Φ ε,n A such that (proof of this result is detailed in appendix B):

Φ ε,n (t) = J Φ ε,n J (t) + A Φ ε,n A (t)
Thanks to this decomposition, one has

∂E ε (t, ũε -ũε,n ) ∂t (t) = ε n Ũ n+1 (t) + εΦ ε,n J (t) T J ∂( Ũ ε -Ũ ε,n ) ∂t (t) + ε n+1 (Φ ε,n A (t)) T A ∂( Ũ ε -Ũ ε,n ) ∂t (t) (107)
Finally, integrating (107) between 0 and T leads to (thanks to the fact that E ε (0, ũεũε,n ) = 0):

E ε (T, ũε -ũε,n ) = T 0 ε n Ũ n+1 (t) + εΦ ε,n J (t) T J ∂( Ũ ε -Ũ ε,n ) ∂t (t)dt + T 0 ε n+1 (Φ ε,n A (t)) T A ∂( Ũ ε -Ũ ε,n ) ∂t (t) (108)

RR n°7265

Using a Green-Riemann formula on the first part of right member in (108) leads to E ε (T, ũεũε,n ) = ε n Ũ n+1 (t) + εΦ ε,n J (t)

T J Ũ ε (t) -Ũ ε,n (t) - T 0 ε n ∂( Ũ n+1 + εΦ ε,n J ) ∂t (t) T J Ũ ε (t) -Ũ ε,n (t) dt + T 0 ε n+1 (Φ ε,n A (t)) T A ∂( Ũ ε -Ũ ε,n ) ∂t (t)dt
(109) By using some Cauchy-Schwartz inequality (allowed even if J and A are not invertible), one gets that, for any vector

V ∈ R N V T J Ũ ε (t) -Ũ ε,n (t) ε 1/2 V T J V 1/2 E ε (t, ũε -ũε,n )(t) V T A ∂( Ũ ε -Ũ ε,n ) ∂t (t) ε -1/2 V T AV 1/2 E ε (t, ũε -ũε,n )(t) (110)
Then, there exists two functions C ε 1 (t) and C ε 2 (t) bounded with respect to ε near 0 such that A.2 Existence and uniqueness of functions W i

E ε (T, ũε -ũε,n ) ε n+ 1 2 C ε 1 (T ) E ε (T, ũε -ũε,n ) + ε n+ 1 2 T 0 C ε 2 (t) E ε (t, ũε -ũε,n )dt (111)
Here, we prove some properties about the functions W i we claimed in section 1.3.2.

Proposition A.2 (recall of the proposition 1.3). There exists a unique W i in H 1 ( J) satisfying (11) and (12)

Proof. The idea is to use the Lax-Milgram theorem on the space

V = V ∈ H 1 ( J) / b J V = 0
On this space, the H 1 semi-norm is equivalent to the H 1 norm, thanks to the Poincare's inequality. We multiply the first line of (11) by a test function V , and we use the Green-Riemann formula:

b J ∇W i ∇V - ∂ b J V (∇W i • n) = 0
We use then second, third and fourth lines, and the equation written above can be rewritten as

b J ∇W i ∇V + j Γj V T j W i = 1 c i Γi W - 1 c i+1 Γi+1 W
Finally, by using definition of T i , we can rewrite this equation as Let us call a(W i , V ) the left part of (114) and ℓ i (V ) the right one. It is an evidence that ℓ i is a continuous linear form on V. Thanks to the Poincare's inequality, a is a bilinear coercive form. For the continuity of this form, just use the proposition A.1 and the fact that the trace operator is continuous from H 1 ( J) to H 1 2 (Γ i ). Finally, using the Lax-Milgram theorem ensure the existence and the uniqueness of W i .

A.3 Properties about the matrix K

We recall here the definition of matrix K ∈ M N -1 (R), given by

K i,j = 1 c j Γj W i - 1 c j+1 Γj+1 W i
Here, we prove another property we claimed in section 1.3.2.

Proposition A.3 (recall of the proposition 1.4). K is a symmetric definite positive matrix

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Proof. The central point of the proof os this proposition is to use the proof of proposition 1.3 given in the previous part of this appendix, in particular equation ( 114) taking V = W j . One can see that we have, thanks to definition of K i,j :

b

J ∇W i ∇W j + N k=1 p∈N * pπ c k Γ k W i w k,p Γ k W j w k,p = K j,i (115)
Equation ( 115) shows immediately that the matrix K is symmetric. Now let us show that K is definite positive. Given U ∈ N N -1 , let us define Φ = N -1 i=1 U i W i . One can see easily that

U t KU = b J |∇Φ| 2 + N i=1 Γi ΦT i Φ b J |∇Φ| 2 0 (116)
We get immediately that K is positive. Then, if U T KU = 0, one can see that Φ = 0, and by looking at Neumann traces on each Γ k , we deduce that U = 0.

B Decomposition on J and A Proposition B.1. Given V ∈ R N , there exist two unique vectors V J and V A such that V = J V J + A V A and J V A = 0 and A V J = 0

Proof. We recall some properties about matrices A and J : these two matrices are symetric positives matrices, are given by ( 15) and (16). We first decompose V on span(1) and its orthogonal:

V = (V • 1) (1 • 1) 1 + V - (V • 1) (1 • 1) 1 (117)
One can see that for any vector W ∈ R N , 1 T J W = 0. Moreover, one has A = | J|(1 • 1) -2 1 1 T . Then by taking

V A = | J| -1 (V • 1)1
one has J V A = 0 and

A V A = | J|(1 • 1) -2 1 1 T | J| -1 (V • 1)1 = (V • 1) (1 • 1) 1
We have partially written V under the form (by using the last relation in (117))

V = A V A + V - (V • 1) (1 • 1) 1 (118)
Now, one has

V - (V • 1) (1 • 1) 1 • 1 = 0
So there exists a vector Φ ∈ R N -1 (that we could even compute by the hand) such that

V - (V • 1) (1 • 1) 1 = P T Φ
Since the matrix K is invertible, we denote by Ψ the unique vector in R N -1 such that Φ = K -1 Ψ. Finally, since P is surjective and its kernel is equal to span(1), there exists a unique vector V J ∈ R N such that Ψ = P V J and 1 • V J = 0. Finally, one gets that

V - (V • 1) (1 • 1) 1 = P T K -1 P V J and A V J = 0
This last result, associated to (118) and ( 16), gives the result of existence. For the uniqueness, let (V J , V A ) and (V ′ J , V ′ A ) be two decompositions such that V = J V J + A V A and J V A = 0 and A V J = 0 V = J V ′ J + A V ′ A and J V ′ A = 0 and A V ′ J = 0

Multiplying each of these two lines by V ′ J -V J and taking the difference leads to, thanks to the last properties on this vectors:

(V ′ J -V J ) T J (V ′ J -V J ) = 0
Denoting Φ = P (V ′ J -V J ) leads to

Φ T K -1 Φ = 0
and thanks to the fact that K is a symmetric definite positive matrix, K -1 is a symmetric definite positive matrix, then Φ = 0; and we deduce that V ′ J -V J = 0. In a same way, since J V A = 0, there exists α ∈ R such that V A = α1; and there exists α ′ ∈ R such that V ′ A = α ′ 1. Multiplying the two lines by 1 and taking the difference leads to

|J | N (α -α ′ ) = 0
This gives the last desired relation.

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Figure 1 :

 1 Figure 1: Configuration of the 1D graph (left) and the ε-thickness 2D graph (right)

Figure 2 :

 2 Figure 2: Configuration of the domain for the general case

c 1 c 2 c 3 Figure 3 :

 233 Figure 3: Configuration of the limit domain (red dotdashed) and the 2D rectanges Ω i (black boxes)

Figure 5 :

 5 Figure 5: Configuration of an example of junction for three slots (in scaled coordinates).

Figure 6 :

 6 Figure 6: Configuration of the new sets Ω ε i,ext and J ε

Figure 8 :

 8 Figure 8: Slots zones Ω ε i (ε) (left figure) and junction zone J ε (ε) (right figure).

Figure 9 :

 9 Figure 9: Configuration of the overlapping domains O i (ε) (grey zones)

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 4 Construction of an improved 1D model 4.1 Construction of Improved Kirchhoff condition

  and P T is continuous injective:

  Corollary 4.1. Given n ∈ N. Let us assume that the terms of the asymptotic expansion may be built up to term n + 2 (this implies thatf ε i ∈ H n+3 (]0, +∞[) and g ε i ∈ H n+2 (]0, +∞[)). Given (δ i > 0) 1 i N , one defines the function ũε,n by ũε,n i (t, s i ) = (t, s i ) on R + ×]εσ i , +∞[Then, there exists a constant C(f, g, t) depending on time and Cauchy data such that, for ε small enough,|∇(ũ εũε,n )(t, s i )| 2 ds i C(f, g, t)ε 2n+2Remark 4.3. One thing that can be remark is that we need the same regularity for both theorems 3.8 and 4.1.

	Finally, using a triangular inequality (as for corollary 3.8) let us allows to express
	the following error estimate:
						n
							ε k ũk
						k=0
	N i=1	∞ δi	∂ ũε -ũε,n ∂t	(t, s i )
	with
	C ε 1 (t) =	Ũ n+1 (t) + εΦ ε,n J (t)
	C ε 2 (t) =	∂( Ũ n+1 + εΦ ε,n J ) ∂t	(t)	T	J	∂( Ũ n+1 + εΦ ε,n J ) ∂t	(t)	(112)
			+ (Φ ε,n A (t))	T A (Φ ε,n A (t))
	using that 2ab a 2 + b 2 let us allow to write that
			E ε (T, ũε -ũε,n )	1 2	ε 2n+1 (C ε 1 (T )) 2 +	1 2	E ε (T, ũε -ũε,n )
						+ ε n+ 1 2	0	T	C ε 2 (t) E ε (t, ũε -ũε,n )dt
	which gives
		1 (T )) 2 E ε (t, ũε -ũε,n )(T ) ε 2n+1 (C ε + 2ε n+ 1 2 T 0 C ε 2 (t) E ε (t, ũε -ũε,n )dt	(113)
									t
									C ε 1 (τ ) +	0	C ε 2 (τ )dτ
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T

J Ũ n+1 (t) + εΦ ε,n J (t)

using back lemma 3.7 gives that that, for any t ∈]0, T [:

E ε (t, ũεũε,n )(t) ε 2n+1 sup τ ∈]0,t[ i 2 +

RR n°7265

This is a particular case. In[START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF], one can see that the authors look for power series expansions with respect to ε and log(ε). These expansions can be even more general.INRIA

L 2 (]0,+∞[) + O(ε n+2 ) (70) INRIA

Acknolowedgement

The authors would like to thanks the people who help somehow to write this paper, especially Katrin Boxberger, Bérangère Delourme, Sébastien Impériale and Xavier Claeys.

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Conclusion

We proved in this report that we are able to give conditions at the "node" of our 1D graph that are more precise that standard Kirchhoff conditions (with respect to the exact solution). One can see that results obtained are a generalization of those obtained in [START_REF] Joly | Construction and analysis of improved kirchhoff conditions for acoustic wave propagation in a junction of thin slots[END_REF], directly in time domain case. One could say that no numerical results are presented here. Indeed, we get the same results as those observed in [START_REF] Joly | Propagation of an acoustic wave in a junction of two thin slots[END_REF] for the time domain case, and there exists some works where we detail the numerical resolution of these type of problems on a more general geometry (with more than one junction, see [START_REF] Semin | Numerical resolution of the wave equation on a network of slots[END_REF]).

A Some properties about the auxiliary problems A.1 Some properties about the DtN map Here, we prove some properties about the DtN maps T i we used. We first recall the definition of T i

2 (]0, c i [) vanishing for constant functions.

Proof. We have several points to prove. To prove the continuity from H

By using some trivial inequalities, one has

Then we use the Cauchy-Schwartz inequality, to obtain

We can recognize product of parts of the H 1 2 (]0, c i [) of functions ϕ and ψ (see [START_REF] Joly | Propagation of an acoustic wave in a junction of two thin slots[END_REF] for more details). This closes proof of the continuity. If we look back the first relation of this proof, which is

we can see that the right member is symmetric (one can invert ϕ and ψ), so we have

Moreover, taking ψ = ϕ shows that the right member of the first relation is positive, which gives < T i ϕ, ϕ > 0. Finally, taking T i = w i,0 leads to


Annick-Camille Dumaret

Dominique-Jeanne Rosset

Adoption and Child Welfare Protection in France