Trapped modes in thin and infinite ladder like domains. Part 2: Asymptotic analysis and numerical application

Abstract

We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a “thick graph”, namely a thin structure (the thinness being characterized by a small parameter $ε > 0$ ) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

Keywords

Spectral theory periodic media quantum graphs matched asymptotic expansion

1. Summary of Part 1 and main results of Part 2

This article is the sequel of [5] and we refer the reader to its introduction for the motivation of the study and related bibliographical comments. We choose to go directly to the heart of the subject and to give below a brief recap about the problem under consideration, then to give a summary of the main results of [5] in Sections 1.1 and 1.2. Finally, we state the main result of the present paper in Section 1.3.

Let $Ω_{ε}$ be a homogeneous periodic domain consisting of the infinite band ${(x, y) \in R \times (- L / 2, L / 2)}$ of height $L > 0$ minus an infinite set of equispaced similar rectangular obstacles (see Fig. 1). The domain $Ω_{ε}$ is 1-periodic with respect to the variable x. The distance between two consecutive obstacles is equal to the distance from the obstacles to the boundary of the band and is denoted by ε.

Fig. 1.

The unperturbed periodic ladder (left). The perturbed ladder (right).

Starting from the periodic domain $Ω_{ε}$ , we introduce a local perturbation by changing the distance between two consecutive obstacles from ε to $μ ε$ , $μ > 0$ (i.e by modifying the size of two consecutive obstacles of $(1 - μ) ε / 2$ ) see Fig. 1 (right) in the case where $μ \in (0, 1)$ ). It corresponds to modify the width of the vertical rung of the ladder from $| x | < ε / 2$ to $| x | < μ ε / 2$ . The corresponding domain is denoted $Ω_{ε}^{μ}$ . We wonder whether such a perturbation creates so called localized modes, that is to say harmonic in time functions of the form $\begin{matrix} (1) & u (x, y, t) = v (x, y) e^{i ω t}, v \in L_{2} (Ω_{ε}), ω \in R, \end{matrix}$ that satisfy the wave equation with homogeneous Neumann boundary condition $\begin{matrix} (2) & \frac{\partial^{2} u}{\partial t^{2}} - Δ u = 0, \partial_{n} u = 0 on \partial Ω_{ε} . \end{matrix}$

1.1. Mathematical formulation of the problem

1.1.1. The operator $A_{ε}^{μ}$

Injecting (1) into (2), one can easily see that the construction of localized modes turns out to solve the following eigenvalue problem for the function v: $\begin{matrix} (3) & - Δ v = ω^{2} v in Ω_{ε}, \partial_{n} v = 0 in \partial Ω_{ε} . \end{matrix}$ It consequently leads us to investigate the spectrum (and more precisely the eigenvalues) of the self-adjoint and positive operator $A_{ε}^{μ}$ , acting in the space $L_{2} (Ω_{ε}^{μ})$ : $\begin{matrix} A_{ε}^{μ} u = - Δ u, D (A_{ε}^{μ}) = {u \in H^{1} (Ω_{ε}^{μ}), Δ u \in L_{2} (Ω_{ε}^{μ}), \partial_{n} u |_{\partial Ω_{ε}^{μ}} = 0} . \end{matrix}$ Based on an asymptotic approach, the spectrum of the operators $A_{ε}^{μ}$ is investigated in [5]. In particular, it is shown that for $μ \in (0, 1)$ , and for $ε > 0$ sufficiently small, the operator $A_{ε}^{μ}$ has eigenvalues. The objective of the present paper is to complement the aforementioned work by constructing a high order asymptotics of these eigenvalues.

1.1.2. The decomposition of the operator $A_{ε}^{μ}$ into its symmetric and antisymmetric components

To study the operator $A_{ε}^{μ}$ , it is convenient to decompose it as the sum of its symmetric and antisymmetric parts. Denoting $L_{2, s} (Ω_{ε}^{μ})$ and $L_{2, a} (Ω_{ε}^{μ})$ the subspaces of $L_{2} (Ω_{ε}^{μ})$ consisting of functions respectively symmetric and antisymmetric (with respect to the axis $y = 0$ ),we have $\begin{matrix} L_{2} (Ω_{ε}^{μ}) = L_{2, s} (Ω_{ε}^{μ}) \oplus L_{2, a} (Ω_{ε}^{μ}) . \end{matrix}$ The operator $A_{ε}^{μ}$ is then decomposed as follows, where $A_{ε, s}^{μ}$ and $A_{ε, a}^{μ}$ are both self-adjoint and positive $\begin{matrix} A_{ε}^{μ} = A_{ε, s}^{μ} \oplus A_{ε, a}^{μ}, with A_{ε, s}^{μ} = A_{ε}^{μ} |_{L_{2, s} (Ω_{ε}^{μ})}, and A_{ε, a}^{μ} = A_{ε}^{μ} |_{L_{2, a} (Ω_{ε}^{μ})} . \end{matrix}$ In this paper, we shall restrict ourselves to the study of the spectrum of the symmetric operator $A_{ε, s}^{μ}$ . Naturally, we could study the antisymmetric one as well.

1.2. The limit problem: Spectral problem on the graph

As might be expected, the investigation of the spectrum of $A_{ε, s}^{μ}$ relies on the investigation of the spectrum of a limit operator denoted by $A_{s}^{μ}$ defined on a graph $G$ obtained as the geometrical limit of the domain $Ω_{ε}^{μ}$ as ε tends to 0. In this section, we define the limit operators $A_{s}^{μ}$ , and we remind important characteristics of its spectrum, established in [5].

1.2.1. The limit operators $A^{μ}$ , $A_{s}^{μ}$ and $A_{a}^{μ}$

The limit periodic graph $G$ . Let us first introduce some notation associated with the limit periodic graph $G = ⋂_{ε > 0} Ω_{ε}^{μ}$ represented on Fig. 2. We denote by $e_{j}$ the vertical edge $e_{j} = {j} \times (- L / 2, L / 2)$ . The edge $e_{0}$ corresponds to the limit of the perturbed rung ${| x | < μ ε / 2}$ . For all j, the upper end of the edge $e_{j}$ is denoted by $M_{j}^{+}$ and the lower one by $M_{j}^{-}$ . The horizontal edge joining $M_{j}^{\pm}$ and $M_{j + 1}^{\pm}$ is denoted by $e_{j + \frac{1}{2}}^{\pm} = (j, j + 1) \times {\pm L / 2}$ .

The sets of all vertices and all edges of the graph are then $\begin{matrix} M = {M_{j}^{\pm}}_{j \in Z}, E = {e_{j}, e_{j + \frac{1}{2}}^{\pm}}_{j \in Z} \end{matrix}$ and we denote by $E (M)$ the set of all the edges of the graph containing the vertex M.

Fig. 2.

Limit graph $G$ .

Weighted functional spaces on $G$ . If u is a function defined on $G$ we will use the following notation $\begin{array}{l} u_{j}^{\pm} = u (M_{j}^{\pm}), u_{j} (y) = u |_{e_{j}}, u_{j + \frac{1}{2}}^{\pm} (x) = u |_{e_{j + \frac{1}{2}}^{\pm}}, \\ (4) & Let w^{μ} : E ⟶ R^{+} such that w^{μ} (e_{0}) = μ, w^{μ} (e) = 1, \forall e \in E, e \neq e_{0} . \end{array}$ Let us now introduce the following weighted functional spaces $\begin{array}{l} (5) & L_{2}^{μ} (G) = {u / u \in L_{2} (e), \forall e \in E; ‖ u ‖_{L_{2}^{μ} (G)}^{2} = \sum_{e \in E} w^{μ} (e) ‖ u ‖_{L_{2} (e)}^{2} < \infty}, \\ (6) & H^{1} (G) = {u \in L_{2}^{μ} (G) / u \in C (G); u \in H^{1} (e), \forall e \in E; ‖ u ‖_{H^{1} (G)}^{2} = \sum_{e \in E} ‖ u ‖_{H^{1} (e)}^{2} < \infty}, \\ (7) & H^{2} (G) = {u \in L_{2}^{μ} (G) / u \in C (G); u \in H^{2} (e), \forall e \in E; ‖ u ‖_{H^{2} (G)}^{2} = \sum_{e \in E} ‖ u ‖_{H^{2} (e)}^{2} < \infty}, \end{array}$ where $C (G)$ denotes the space of continuous functions on $G$ .

Definition of the limit operators $A^{μ}$ , $A_{s}^{μ}$ and $A_{a}^{μ}$ . As known since the work of [4,7,18], the limit operator $A^{μ}$ acting on $L_{2}^{μ} (G)$ is defined as follows: denoting by $u_{e}$ the restriction of u to e, $\begin{matrix} (8) & \begin{array}{l} {(A^{μ} u)}_{e} = - u_{e}^{″}, \forall e \in E, \\ D (A^{μ}) = {u \in H^{2} (G) / \sum_{e \in E (M)} w^{μ} (e) u_{e}^{'} (M) = 0, \forall M \in M}, \end{array} \end{matrix}$ where $u_{e}^{'} (M)$ stands for the derivative of the function $u_{e}$ at the point M in the outgoing direction. The vertex relations in (8) are called Kirchhoff’s conditions. Note that they all have an identical expression except at the vertices $M_{0}^{\pm}$ . We remark that the perturbation, which results from a geometrical modification of the domain for the initial operator, is taken into account at the limit by means of the Kirchhoff’s conditions at the vertices $M_{0}^{\pm}$ . The operator $A^{μ}$ is indeed positive and self-adjoint (cf. [17, Section 3.3]).

As for the ladder, denoting $L_{2, s}^{μ} (G)$ and $L_{2, a}^{μ} (G)$ the subspaces of $L_{2}^{μ} (G)$ consisting of functions respectively symmetric and antisymmetric (with respect to the axis $y = 0$ ) we have $\begin{matrix} L_{2}^{μ} (G) = L_{2, s}^{μ} (G) \oplus L_{2, a}^{μ} (G) . \end{matrix}$ Thus, the operator $A^{μ}$ can be decomposed into the orthogonal sum where $A_{s}^{μ}$ and $A_{a}^{μ}$ are again self-adjoint positive operators $\begin{matrix} A^{μ} = A_{s}^{μ} \oplus A_{a}^{μ}, with A_{s}^{μ} = A^{μ} |_{L_{2, s}^{μ} (G)}, and A_{a}^{μ} = A^{μ} |_{L_{2, a}^{μ} (G)} . \end{matrix}$

1.2.2. The spectrum of

A_{s}^{μ}

The operator $A_{s}^{μ}$ being self-adjoint, its spectrum consists of its essential spectrum $σ_{ess} (A_{s}^{μ})$ and its discrete spectrum $σ_{d} (A_{s}^{μ})$ . It is well-known that $σ_{ess} (A_{s}^{μ})$ coincides with the spectrum of the periodic operator $A_{s} : = A_{s}^{μ}$ for $μ = 1$ (see e.g. [3, Theorem 4, Chapter 9]) and has a band-gap structure [6,16,26]: $\begin{matrix} σ_{ess} (A_{s}^{μ}) = σ (A_{s}) = ⋃_{n \in N} [a_{n}, b_{n}], \end{matrix}$ where $a_{0} ⩾ 0$ , $a_{n} < a_{n + 1}$ and $a_{n} ⩽ b_{n}$ . In general, the segments $[a_{n}, b_{n}]$ may overlap. If, for some $n \in N$ , $b_{n} < a_{n + 1}$ , the open interval $] b_{n}, a_{n + 1} [$ is called a gap of the essential spectrum operator $A_{s}^{μ}$ .

The following proposition, based on explicit characterizations of $σ_{ess} (A_{s}^{μ})$ and $σ_{d} (A_{s}^{μ})$ , is proved in [5, Proposition 5 and Theorem 1]:

Proposition 1.

The essential spectrum of the operator $A_{s}^{μ}$ has infinitely many gaps whose ends tend to infinity.

For $μ ⩾ 1$ , the discrete spectrum of the operator $A_{s}^{μ}$ is empty, while for $μ \in (0, 1)$ , the operator $A_{s}^{μ}$ has exactly one or two eigenvalue(s) in each of its gaps, each eigenvalue being simple.

To show the last item, we use the characterization of the spectrum $\begin{array}{l} (9) & λ \in σ_{d} (A_{s}^{μ}) \Leftrightarrow | g (\sqrt{λ}) | > 1 and μ = 1 - \sqrt{\frac{g^{2} (\sqrt{λ}) - 1}{{(g (\sqrt{λ}) + cos \sqrt{λ})}^{2}}} \\ (10) & where \forall ω > 0, g (ω) = - cos (ω) + \frac{sin (ω) tan (ω L / 2)}{2} . \end{array}$ An associated eigenvector u is given, on the horizontal edges $e_{j + 1 / 2}^{\pm}$ ( $j \in Z$ ), by $\begin{array}{l} (11) & u_{j + \frac{1}{2}} (s) = r {(\sqrt{λ})}^{| j |} (sin (\sqrt{λ} (1 - s)) + r (\sqrt{λ}) sin (\sqrt{λ} s)) / sin \sqrt{λ}, s : = x - j \in [0, 1], \end{array}$ while, on the vertical edges $e_{j}$ ( $j \in Z$ ), it is given by $\begin{array}{l} (12) & u_{j} (y) = r {(\sqrt{λ})}^{| j |} cos (\sqrt{λ} y) / cos (\sqrt{λ} L / 2), y \in [- L / 2, L / 2], \end{array}$ where $r (\sqrt{λ})$ is the unique root in $(- 1, 1)$ of the following characteristic equation $\begin{matrix} (13) & r^{2} + 2 g (\sqrt{λ}) r + 1 = 0 . \end{matrix}$
1.3. Main result

Before stating the main result of this paper, let us remind first the result, already proven for instance in [18], which states the convergence of the essential spectrum of the operator $A_{ε, s}^{μ}$ to the essential spectrum of $A_{s}^{μ}$ . More precisely, let $(a, b)$ be a gap of the operator $A_{s}^{μ}$ on the limit graph $G$ then, there exists $ε_{0} > 0$ such that if $ε < ε_{0}$ the operator $A_{ε, s}^{μ}$ has a gap $(a^{ε}, b^{ε})$ whose extremities satisfy $\begin{matrix} (14) & a^{ε} = a + O (ε) and b^{ε} = b + O (ε) . \end{matrix}$ This means that for some $C_{1} > 0$ and for any $ε < ε_{0}$ $\begin{matrix} (15) & σ_{ess} (A_{ε, s}^{μ}) \cap [a + C_{1} ε, b - C_{1} ε] = \emptyset . \end{matrix}$ In [5, Theorem 1], we have shown that for $μ \in (0, 1)$ , and for $ε > 0$ sufficiently small the discrete spectrum of $A_{ε, s}^{μ}$ is not empty. More precisely, for $μ \in (0, 1)$ , the discrete spectrum of $A_{s}^{μ}$ is not empty and if $λ^{(0)} \in (a, b)$ is an eigenvalue of this operator, then there exists $0 < ε_{1} < ε_{0}$ such that if $ε < ε_{1}$ the operator $A_{ε, s}^{μ}$ has an eigenvalue $λ^{ε}$ inside the gap $(a^{ε}, b^{ε})$ . Moreover, $\begin{matrix} (16) & λ^{ε} = λ^{(0)} + O (\sqrt{ε}) . \end{matrix}$ The present paper complements the result of [5] by obtaining and constructing an asymptotic expansion of these eigenvalues at any order with respect to the parameter ε.

Theorem 1.
Let $λ^{(0)} \in (a, b)$ be an eigenvalue of the operator $A_{s}^{μ}$ and for ε small enough, $λ^{ε}$ the unique eigenvalue of the operator $A_{ε, s}^{μ}$ satisfying ( 16 ). Then, there exists a real sequence ${(λ^{(k)})}_{k \in N^{}}$ , which is constructed inductively with the help of an algorithm that is presented in detail in Section* 5 , such that $\begin{matrix} (17) & \forall n \in N, λ^{ε} = \sum_{k = 0}^{n} ε^{k} λ^{(k)} + O (ε^{n + 1}) . \end{matrix}$
Remark 1.
Note that as every eigenvalue of the operators $A_{s}^{μ}$ is simple (as established in Prop. 1), for ε small enough, $λ^{ε}$ is a simple eigenvalue of $A_{ε, s}^{μ}$ , see [25]. As soon as one obtains such asymptotic expansions for these simple eigenvalues, it is also possible to deduce an asymptotic expansion for a prescribed associated eigenvector (see for instance [19, Part 4]).

We point out that the main novelty of the previous result lies in the determination of a high order asymptotic expansion of the eigenvalues of $A_{ε, s}^{μ}$ , and we use it for numerical computations. The convergence of the spectrum of $A_{ε, s}^{μ}$ toward the spectrum of $A_{s}^{μ}$ has been proved in [25]. The case of ‘fattened’ compact graphs with more general boundary conditions has been investigated in [10] while the specific case of Dirichlet boundary conditions was also studied in [21]. A more complete presentation of related bibliographical references can be found in [5].
2. Methodology of the proof and asymptotic expansion ansatz

2.1. Methodology of the proof

The proof of Theorem 1 relies on the following lemma.

Lemma 1.
Let $λ^{(0)} \in (a, b)$ be an eigenvalue of $A_{s}^{μ}$ . Suppose that there exist two sequences of real numbers ${(λ^{(m)})}_{m \in N^{}}$ and* ${(α_{m})}_{m \in N^{}}$ , with* $\begin{matrix} (18) & 0 < α_{m} ⩽ m + 1 and lim_{m \to + \infty} α_{m} = + \infty \end{matrix}$ and a sequence $u^{ε, m} \in H_{s}^{1} (Ω_{ε}^{μ})$ such that, for any $v \in H_{s}^{1} (Ω_{ε}^{μ})$ $\begin{matrix} (19) & | \int_{Ω_{ε}^{μ}} (\nabla u^{ε, m} \nabla v - λ^{ε, m} u^{ε, m} v) | ⩽ C ε^{α_{m}} {‖ u^{ε, m} ‖}_{H^{1} (Ω_{ε}^{μ})} ‖ v ‖_{H^{1} (Ω_{ε}^{μ})}, where λ^{ε, m} = \sum_{k = 0}^{m} ε^{k} λ^{(k)} . \end{matrix}$ Then there exists at least one eigenvalue $λ^{ε}$ of $A_{ε, s}^{μ}$ such that $\begin{matrix} \forall n \in N, λ^{ε} = \sum_{k = 0}^{n} ε^{k} λ^{(k)} + O (ε^{n + 1}) . \end{matrix}$
Proof.
See the electronic version. □

The function $u^{ε, m}$ is called a pseudo-mode. This pseudo-mode and the associated expansion $λ^{ε, m}$ are constructed thanks to a formal asymptotic expansion of an eigenpair ( $u^{ε}$ , $λ^{ε}$ ) of the following problem $\begin{matrix} (20) & |\begin{matrix} Find u^{ε} \in H^{1} (Ω_{ε}^{μ}) s.t. u^{ε} (x, y) = u^{ε} (x, - y), \forall (x, y) \in Ω_{ε}^{μ}, u^{ε} \neq 0, and λ^{ε} \in R^{+}, \\ Δ u^{ε} + λ^{ε} u^{ε} = 0 in Ω_{ε}^{μ}, \partial_{n} u^{ε} = 0 on \partial Ω_{ε}^{μ} . \end{matrix} \end{matrix}$ This asymptotic expansion will be constructed by induction starting from an eigenvalue $λ^{(0)} \in (a, b)$ of the operator $A_{s}^{μ}$ and an associated eigenvector $u^{(0)}$ .

Due to the multiscale nature of the problem, it is not possible to construct a simple asymptotic expansion of $u^{ε}$ that would be valid in the whole domain $Ω_{ε}^{μ}$ . We need to distinguish two asymptotic expansions of $u^{ε}$ . The first one, describing the overall behaviour of $u^{ε}$ far from the junctions, is expressed by means of the longitudinal coordinate s ( $s = x - j$ for the j-th horizontal thin slit and $s = y$ for the vertical thin slits) and is called the far field expansion. The second one is the near field expansion and is used to approximate $u^{ε}$ in the neighborhood of each junction. Thus, it is expressed by means of the fast variables $((x - j) / ε, (y + L / 2) / ε)$ near the j-th junction and is defined on a normalized unperturbed junction for $j \neq 0$ and a normalized perturbed junction for $j = 0$ . Since both expansions are meant to be two approximations of the same function $u^{ε}$ , they have to satisfy some matching conditions in some intermediate zones. This method is often called Matched Asymptotic Expansion. For complete and detailed descriptions of the method, we refer the reader to [11,28] and [19] (cf. Part IV dedicated to eigenvalue problems). See also [2] for a recent application of the method to a spectral problem. See also [23,24] and [1, Chapter 8] for another asymptotic study of similar periodic skeletal structures.

In Section 2.2, we give the ansatz for the far field and the near field expansions as sums of far field and near field terms indexed by $n \in N$ and derive the problems (defined inductively on n) satisfied by these far field and near field terms. We give the matching conditions in Section 2.3. We study in Sections 3.1–3.2 and in Section 3.3 the well posedness of the problems satisfied by the near field terms and the far field terms respectively. We explain the algorithm of construction of each term in Section 4 and finally prove Theorem 1 by establishing (19) in Section 5.

Fig. 3.
The domain $Ω_{ε}^{μ, -}$ .

Before entering the details, let us introduce some notations. The function $u^{ε}$ being even in y, it suffices to construct an asymptotic expansion of $u^{ε}$ on the lower half part $Ω_{ε}^{μ, -}$ of $Ω_{ε}^{μ}$ (comb shape domain): $\begin{matrix} Ω_{ε}^{μ, -} = {(x, y) \in Ω_{ε}^{μ} s.t. y < 0} . \end{matrix}$ As represented on Fig. 3, we denote by $E_{j + \frac{1}{2}}^{ε, -}$ , $j \in Z$ , the horizontal slits of the domain $Ω_{ε}^{μ, -}$ $\begin{array}{l} E_{j + \frac{1}{2}}^{ε, -} = (j + ε μ_{j} / 2, (j + 1) - ε μ_{j + 1} / 2) \times (- L / 2, - L / 2 + ε), \end{array}$ by $E_{j}^{ε, -}$ its vertical slits $E_{j}^{ε, -} = (j - ε μ_{j} / 2, j + ε μ_{j} / 2) \times (- L / 2 + ε, 0)$ .

The domains $K_{j}^{ε, -}$ are the junctions $K_{j}^{ε, -} = (j - ε μ_{j} / 2, j + ε μ_{j} / 2) \times (- L / 2, - L / 2 + ε)$ where for all $j \in Z$ , $μ_{j} = 1$ if $j \neq 0$ and $μ_{0} = μ$ .
2.2. Asymptotic expansions: Ansatz and equations

From now on, $λ^{(0)} \in (a, b)$ is an eigenvalue of $A_{s}^{μ}$ and $u^{(0)}$ an associated eigenvector: $\begin{matrix} (21) & u^{(0)} \in D (A_{s}^{μ}), A_{s}^{μ} u^{(0)} = λ^{(0)} u^{(0)} \end{matrix}$ i.e. denoting $\forall j \in Z$ $u_{j + \frac{1}{2}}^{(0)} \equiv u^{(0)} |_{e_{j + 1 / 2}^{\pm}}$ and $u_{j}^{(0)} \equiv u^{(0)} |_{e_{j}}$ (using the notations of Section 1.2.1) $\begin{matrix} (22) & \forall j \in Z, \{\begin{matrix} \partial_{s}^{2} u_{j + \frac{1}{2}}^{(0)} (s) + λ^{(0)} u_{j + \frac{1}{2}}^{(k)} (s) = 0, s = x - j \in [0, 1], \\ \partial_{y}^{2} u_{j}^{(0)} (y) + λ^{(0)} u_{j}^{(0)} (y) = 0, y \in [- L / 2, 0], \\ \partial_{y} u_{j}^{(0)} (0) = 0, \\ u_{j - \frac{1}{2}}^{(0)} (1) = u_{j}^{(0)} (- L / 2) = u_{j + \frac{1}{2}}^{(0)} (0), \\ \partial_{s} u_{j + \frac{1}{2}}^{(0)} (0) - \partial_{s} u_{j - \frac{1}{2}}^{(0)} (1) + μ_{j} \partial_{y} u_{j}^{(0)} (- L / 2) = 0 \end{matrix} \end{matrix}$ where we denote $\partial_{s}$ (resp. $\partial_{y}$ ) the derivative with respect to s (resp. to y).

To start the construction of the asymptotic expansion, we have to fix $u^{(0)}$ . We have chosen $u^{(0)}$ such that $u_{0}^{(0)} (- L / 2) = 1$ , namely, $\forall j \in Z$ , $\begin{array}{l} (23) & u_{j + \frac{1}{2}}^{(0)} (s) = r^{| j |} \frac{sin (\sqrt{λ^{(0)}} (1 - s))}{sin \sqrt{λ^{(0)}}} + r^{| j + 1 |} \frac{sin (\sqrt{λ^{(0)}} s)}{sin \sqrt{λ^{(0)}}}, s = x - j \in [0, 1], \\ (24) & u_{j}^{(0)} (y) = r^{| j |} \frac{cos (\sqrt{λ^{(0)}} y)}{cos (\sqrt{λ^{(0)}} L / 2)}, y \in [- L / 2, L / 2] . \end{array}$ Here r stands for $r (\sqrt{λ^{(0)}})$ defined in (13). Note that $u^{(0)}$ is exponentially decaying with $| x |$ .

Remark 2.
Choosing another eigenvector $u^{(0)}$ leads obviously to the asymptotic expansion of a different $u_{ε}$ but this does not change the asymptotic expansion of $λ^{ε}$ (see Remark 7).

We propose an asymptotic expansion for $λ^{ε}$ and $u^{ε}$ solution of (20) constructed by induction starting from $λ^{(0)}$ and $u^{(0)}$ . In the following, $O (ε^{\infty})$ will always denote a remainder that (formally) decays more rapidly that any power of ε. We first suppose a formal power series expansion for the eigenvalue: $\begin{matrix} (25) & λ^{ε} = \sum_{k \in N} ε^{k} λ^{(k)} + O (ε^{\infty}) . \end{matrix}$
Remark 3.
Throughout the rest of the paper, we shall extend the above convention: any quantity indexed by k with $k < 0$ is automatically 0.

Fig. 4.
Schematic representation of the asymptotic expansion (NF: near field, FF: far field).

Mimicking the approach of [13,14,27], we use the following ansatz (see Fig. 4)

Far field asymptotic expansion: in the horizontal thin slits $E_{j + \frac{1}{2}}^{ε, -}$ , $j \in Z$ , we assume that $\begin{array}{l} (26) & u^{ε} (x, y) \approx u_{j + \frac{1}{2}}^{ε} (x, y) = \sum_{k \in N} ε^{k} u_{j + \frac{1}{2}}^{(k)} (s) + O (ε^{\infty}), s = x - j, (x, y) \in E_{j + \frac{1}{2}}^{ε, -} . \end{array}$ In the same way, in the vertical thin slits $E_{j}^{ε, -}$ , $j \in Z$ , we assume that $\begin{array}{l} (27) & u^{ε} (x, y) \approx u_{j}^{ε} (x, y) = \sum_{k \in N} ε^{k} u_{j}^{(k)} (y) + O (ε^{\infty}), (x, y) \in E_{j}^{ε, -} . \end{array}$ With the ansatz (26) and (27), we anticipate that, for small ε, in each slit, the field is essentially a 1D field in the longitudinal variable, except maybe close to the junctions, the transverse variations being contained in the $O (ε^{\infty})$ remainders. The 1D functions appearing in the above expansions are defined on the edges of the half graph $G^{-} = G \cap {y < 0}$ . More precisely $\begin{matrix} u_{j + \frac{1}{2}}^{(k)} : e_{j + \frac{1}{2}}^{-} \equiv [0, 1] \mapsto R, u_{j}^{(k)} : e_{j}^{-} \equiv [- L / 2, 0] \mapsto R . \end{matrix}$ These functions are independent of ε and given by (23–24) for $k = 0$ . In what follows, for any k, collecting these functions over the index j, we define a function $u^{(k)} : G^{-} \mapsto R$ (by definition the far field of order k) $\begin{matrix} (28) & u^{(k)} \in H_{br}^{1} (G^{-}), u^{(k)} = u_{j + \frac{1}{2}}^{(k)} on e_{j + \frac{1}{2}}^{-}, u^{(k)} = u_{j}^{(k)} on e_{j}^{-}, \end{matrix}$ where $H_{br}^{1} (G^{-})$ is the Hilbert space $\begin{matrix} (29) & H_{br}^{1} (G^{-}) = {u \in L_{2}^{μ} (G^{-}) / u \in H^{1} (e), \forall e \in E; \sum_{e \in E} ‖ u ‖_{H^{1} (e)}^{2} < \infty} . \end{matrix}$ If we denote $H^{1} (G^{-}) = {u |_{G^{-}}, u \in H^{1} (G)}$ , we have $H^{1} (G^{-}) \subset H_{br}^{1} (G^{-})$ : the larger space differs from the smaller one by the fact that we removed the continuity condition (see the definition (6) of $H^{1} (G)$ . As we shall see, except for $k = 0$ , all $u^{(k)}$ will be discontinuous on $G$ .

Substituting (25–26–27) into the eigenvalue problem (20), and separating formally the different powers of ε, we get the following set of problems for the far field terms $u^{(k)}$ , $k \in N^{}$ : $\forall j \in Z$ , $\begin{array}{rcl} (30) & \{\begin{matrix} \partial_{s}^{2} u_{j + \frac{1}{2}}^{(k)} (s) + λ^{(0)} u_{j + \frac{1}{2}}^{(k)} (s) = - \sum_{m = 0}^{k - 1} λ^{(k - m)} u_{j + \frac{1}{2}}^{(m)} (s), s \in (0, 1), \\ \partial_{y}^{2} u_{j}^{(k)} (y) + λ^{(0)} u_{j}^{(k)} (y) = - \sum_{m = 0}^{k - 1} λ^{(k - m)} u_{j}^{(m)} (y), y \in (- L / 2, 0), \partial_{y} u_{j}^{(k)} (0) = 0 . \end{matrix} \end{array}$ The far field terms $u^{(k)}$ are still not completely defined by (30). More precisely, we need to prescribe transmission conditions at each node $M_{j}$ of the graph $G$ to link the functions $\begin{matrix} (u_{j + \frac{1}{2}}^{(k)}, u_{j - \frac{1}{2}}^{(k)}, u_{j}^{(k)}) . \end{matrix}$ These transmission conditions will result from the so-called matching conditions.

Near field asymptotic expansion: in the neighborhood of the junctions $K_{j}^{ε, -}$ , $j \in Z$ , we assume that the following expansion holds $\begin{array}{l} (31) & u^{ε} (x, y) \approx U_{j}^{ε} (x, y) = \sum_{k \in N} ε^{k} U_{j}^{(k)} (\frac{x - j}{ε}, \frac{y + L / 2}{ε}), \end{array}$ where the near field terms $U_{j}^{(k)}$ do not depend on ε and are defined on “infinite T-shaped junctions”: $\begin{matrix} U_{j}^{(k)} : J_{j} \mapsto R, j \in Z, \end{matrix}$ where the junctions $J_{j}$ (all identical for $j \neq 0$ ) are defined by (see also Fig. 5) $\begin{matrix} (32) & |\begin{matrix} J_{j} = K_{j} \cup B_{j, -} \cup B_{j, +} \cup B_{j, 0}, K_{j} = [- \frac{μ_{j}}{2}, \frac{μ_{j}}{2}] \times [0, 1], \\ B_{j, \pm} = \pm (\frac{μ_{j}}{2}, + \infty) \times (0, 1), B_{j, 0} = (- \frac{μ_{j}}{2}, \frac{μ_{j}}{2}) \times (1, + \infty) . \end{matrix} \end{matrix}$

In what follows, for any k, we denote by $U^{(k)}$ (near field of order k) the set of the near field functions $\begin{matrix} (33) & U^{(k)} : = {U_{j}^{(k)}}_{j \in Z} . \end{matrix}$ As usual (see [13,14]), we look for near field terms that belong to $\begin{matrix} (34) & \{\begin{matrix} U_{j}^{(k)} \in V_{j} : = {U \in H_{loc}^{1} (J_{j}), w_{j} U \in H^{1} (J_{j})} where \\ w_{j} (X, Y) = 1 in K_{j}, e^{- \sqrt{\pm (X \mp μ_{j} / 2)}} in B_{j, \pm}, e^{- \sqrt{Y - 1}} in B_{j, 0} \end{matrix} \end{matrix}$ which prevents an exponential growth but authorizes a polynomial one. Note that, for $j \neq 0$ , $J_{j}$ can be identified to $J_{1}$ , $V_{j}$ can be identified to $V_{1}$ . Substituting (25–31) into (20) and separating formally the different powers of ε, we find a set of problems for the near field functions $U^{(k)}$ , $k \in N$ : $\begin{matrix} (35) & \forall j \in Z, \{\begin{matrix} Δ U_{j}^{(k)} = - \sum_{m = 0}^{k - 2} λ^{(k - m - 2)} U_{j}^{(m)} in J_{j}, & (i) \\ \partial_{n} U_{j}^{(k)} = 0 on \partial J_{j} . & (ii) \end{matrix} \end{matrix}$ As for the far field terms, near field terms $U^{(k)}$ are not completely defined by (35) (for instance, any constant function satisfies Problem (35) for $k ⩽ 1$ or more generally any harmonic function that belongs to $V_{j}$ ). We need to prescribe their behavior at infinity (in the three infinite branches $B_{j, δ}$ , $δ \in {+, -, 0}$ ), which here again results from the matching conditions.
Fig. 5.
The junctions $J_{j}$ (perturbed for $j = 0$ (left) and unperturbed for $j \neq 0$ (right)).

To make the matching conditions more explicit, we need to describe the form of the near field terms in the three infinite branches $B_{j, δ}$ of $J_{j}$ , which relies on a modal expansion. To do so, for any $δ \in {0, +, -}$ , we denote by $(t, s)$ the transversal and longitudinal variables of $B_{j, δ}$ , i.e. $\begin{matrix} t = \{\begin{matrix} X & in B_{j, 0}, \\ Y & in B_{j, \pm}, \end{matrix} s = \{\begin{matrix} Y & in B_{j, 0}, \\ X & in B_{j, \pm} \end{matrix} \end{matrix}$ and we introduce the two following bases of $L^{2} (- μ_{j} / 2, μ_{j} / 2)$ and $L^{2} (- 1 / 2, 1 / 2)$ and $\begin{matrix} (36) & v_{ℓ, 0}^{j} (t) : = \sqrt{\frac{2}{μ_{j}}} cos (\frac{ℓ π}{μ_{j}} (t + μ_{j} / 2)) and v_{ℓ, \pm}^{j} (t) = v_{ℓ} (t) : = \sqrt{2} cos (ℓ π t), \end{matrix}$ which are nothing but the eigenfunctions of the corresponding 1D Neumann laplacians.

We begin with some recaps about solution of homogeneous Laplace equations in bands. More precisely, we are interested in the problem $\begin{matrix} (37) & - Δ u_{j}^{δ} (φ) = 0 in B_{j, δ}, u_{j}^{δ} (φ) = φ on Σ_{j, δ}, \partial_{n} u_{j}^{δ} (φ) = 0 on \partial B_{j, δ} ∖ Σ_{j, δ} \end{matrix}$ where φ is the Dirichlet data belonging to $H^{\frac{1}{2}} (Σ_{j, δ})$ , with $δ \in {0, +, -}$ . It is well-known that this problem has a unique solution in the space $\begin{array}{l} (38) & H^{1} (B_{j, δ}) : = {v \in H_{loc}^{1} (B_{j, δ}), \frac{v}{\sqrt{1 + s^{2}}} \in L^{2} (B_{j, δ}), \nabla v \in L^{2} (B_{j, δ})} \supset H^{1} (B_{j, δ}) \\ (39) & and defining H arm (B_{δ, j}) : = {U \in H^{1} (B_{δ, j}), Δ U = 0 in B_{δ, j}, \partial_{n} U = 0 on \partial B_{δ, j} ∖ Σ_{δ, j}}, \\ (40) & the mapping φ \mapsto u_{j}^{δ} (φ) is an isomorphism from H^{\frac{1}{2}} (Σ_{j, δ}) into the space H arm (B_{δ, j}) . \end{array}$

The well-posedness of (37) follows from Lax–Milgram’s lemma and Hardy’s inequality (see for instance [22, Lemma 2.5.7]). Moreover, it can be solved by separation of variables in $(s, t)$ , yielding $\begin{matrix} (41) & \{\begin{matrix} u_{j}^{\pm} (φ) = \sum_{ℓ = 0}^{+ \infty} β_{ℓ, \pm} e^{- ℓ π | s |} v_{ℓ} (t) in B_{j, \pm}, β_{ℓ, \pm} = {(φ, v_{ℓ})}_{L^{2} (Σ_{j, \pm})}, \\ u_{j}^{0} (φ) = \sum_{ℓ = 0}^{+ \infty} β_{ℓ, 0} e^{- \frac{ℓ π}{μ_{j}} | s |} v_{ℓ, 0}^{j} (t) in B_{j, 0}, β_{ℓ, 0} = {(φ, v_{ℓ, 0}^{j})}_{L^{2} (Σ_{j, 0})}, \end{matrix} \end{matrix}$ where, we see that $u_{j}^{δ}$ tends exponentially fast to a constant as $| s |$ tends to $+ \infty$ . We are now ready to give the structure of the near field terms in the bands $B_{j, δ}$ . Proposition 2.
Assume that there exists a sequence of near fields* ${U^{(k)}, k ⩾ 0}$ (see ( 33 )), with $U_{j}^{(k)} \in V_{j}$ , $\forall j \in Z$ , satisfying ( 35 ). Then, for any $k \in N$ , for any $j \in Z$ , there exists 1D polynomials $\begin{matrix} ({\hat{U}}_{j, \pm}^{(k)}, {\hat{U}}_{j, 0}^{(k)}) \in P_{k + 1} and {({\hat{U}}_{j, ℓ, \pm}^{(k)}, {\hat{U}}_{j, ℓ, 0}^{(k)}) \in P_{[k / 2]}, ℓ ⩾ 1} \end{matrix}$ such that the function $U_{j}^{(k)}$ admits the following decomposition in the bands $B_{j, δ}$ $\begin{matrix} (42) & \{\begin{matrix} U_{j}^{(k)} = {\hat{U}}_{j, \pm}^{(k)} (s) + \sum_{ℓ = 1}^{+ \infty} {\hat{U}}_{j, ℓ, \pm}^{(k)} (s) e^{- ℓ π | s |} v_{ℓ} (t) in B_{j, \pm}, \\ U_{j}^{(k)} = {\hat{U}}_{j, 0}^{(k)} (s) + \sum_{ℓ = 1}^{+ \infty} {\hat{U}}_{j, ℓ, 0}^{(k)} (s) e^{- \frac{ℓ π}{μ_{j}} | s |} v_{ℓ, 0}^{j} (t) in B_{j, 0} \end{matrix} \end{matrix}$ where the above series converge in $H_{loc}^{1} (B_{j, δ})$ . A more detailed description of $U_{j}^{(k)}$ in each band is $\begin{matrix} (43) & U_{j}^{(k)} = α_{j, δ}^{(k)} s + U_{j, δ}^{(k)} (t, s) + U_{j, δ, part}^{(k - 2)} (t, s) in B_{j, δ} \end{matrix}$ where $α_{j, δ}^{(k)}$ is a real constant and
$U_{j, δ}^{(k)}$ belongs to $H arm (B_{j, δ})$ (see Def. ( 39 ), in particular $Δ U_{j, δ}^{(k)} = 0$ ), thus of the form (see ( 41 )) $\begin{matrix} (44) & \{\begin{matrix} U_{j, \pm}^{(k)} = β_{j, \pm}^{(k)} + \sum_{ℓ = 1}^{+ \infty} β_{j, ℓ, \pm}^{(k)} e^{- ℓ π | s |} v_{ℓ} (t) in B_{j, \pm}, \\ U_{j, 0}^{(k)} = β_{j, 0}^{(k)} + \sum_{ℓ = 1}^{+ \infty} β_{j, ℓ, 0}^{(k)} e^{- \frac{ℓ π}{μ_{j}} | s |} v_{ℓ, 0}^{j} (t) in B_{j, 0} . \end{matrix} \end{matrix}$

$U_{j, δ, part}^{(k - 2)} \in H_{loc}^{1} (B_{j, δ})$ is the only particular solution of $\begin{matrix} (45) & Δ U_{j, δ, part}^{(k - 2)} = - \sum_{m = 0}^{k - 2} λ^{(k - m - 2)} U_{j}^{(m)} in B_{j, δ}, \partial_{n} U_{j, δ, part}^{(k - 2)} = 0 on \partial B_{δ, j} ∖ Σ_{δ, j} \end{matrix}$ that admits a decomposition of the form: $\begin{array}{l} (46) & \{\begin{matrix} U_{j, \pm, part}^{(k - 2)} = {\hat{U}}_{j, \pm, part}^{(k - 2)} (s) + \sum_{ℓ = 1}^{+ \infty} {\hat{U}}_{j, ℓ, \pm, part}^{(k - 2)} (s) e^{- ℓ π | s |} v_{ℓ} (t), \\ U_{j, 0, part}^{(k - 2)} = {\hat{U}}_{j, 0, part}^{(k - 2)} (s) + \sum_{ℓ = 1}^{+ \infty} {\hat{U}}_{j, ℓ, 0, part}^{(k - 2)} (s) e^{- \frac{ℓ π}{μ_{j}} | s |} v_{ℓ, 0}^{j} (t), \end{matrix} \\ (47) & with \{\begin{matrix} {\hat{U}}_{j, δ, part}^{(k - 2)} \in P_{k} & and satisfies (a) {\hat{U}}_{j, δ, part}^{(k - 2)} (0) = \partial_{s} {\hat{U}}_{j, δ, part}^{(k - 2)} (0) = 0, \\ {\hat{U}}_{j, ℓ, δ, part}^{(k - 2)} \in P_{[k / 2]} & and satisfies (b) {\hat{U}}_{j, ℓ, δ, part}^{(k - 2)} (0) = 0 . \end{matrix} \end{array}$

Remark 4.
The link between (42) and (43–46) is quite straightforward. In particular, $\begin{matrix} (48) & {\hat{U}}_{j, δ}^{(k)} (s) = α_{j, δ}^{(k)} s + β_{j, δ}^{(k)} + {\hat{U}}_{j, δ, part}^{(k - 2)} (s), {\hat{U}}_{j, ℓ, δ}^{(k)} (s) = β_{j, ℓ, δ}^{(k)} + {\hat{U}}_{j, ℓ, δ, part}^{(k - 2)} (s) . \end{matrix}$ In the decomposition (43), the label $(k - 2)$ for the functions $\begin{matrix} U_{j, δ, part}^{(k - 2)}, {\hat{U}}_{j, δ, part}^{(k - 2)} or {\hat{U}}_{j, ℓ, δ, part}^{(k - 2)} \end{matrix}$ indicates that they are entirely determined by $λ^{(m)}$ and $U_{j}^{(m)}$ for $m ⩽ k - 2$ .

In opposition, “the harmonic part” of $U_{j}^{(k)}$ in the band $B_{j, δ}$ , namely the function $\begin{matrix} α_{j, δ}^{(k)} s + U_{j, δ}^{(k)} (t, s), \end{matrix}$ involves infinitely many “free parameters” $(α_{j, δ}^{(k)}, β_{j, δ}^{(k)})$ and ${β_{j, ℓ, δ}^{(k)}, ℓ ⩾ 1}$ . In the following, the decomposition (42) will be used for deriving the matching conditions in Section 2.3. On the other hand, we shall prefer to use the alternate formula (43) for the construction of the terms of the asymptotic expansion (see in particular Section 3).
Remark 5.
Integrating (46) with respect to $t$ gives $\begin{array}{l} \int_{0}^{1} U_{j, \pm, part}^{(k - 2)} (s, t) d t = {\hat{U}}_{j, \pm, part}^{(k - 2)} (s) and \int_{- \frac{μ_{j}}{2}}^{\frac{μ_{j}}{2}} U_{j, 0, part}^{(k - 2)} (s, t) d t = μ_{j} {\hat{U}}_{j, 0, part}^{(k - 2)} (s) \\ as well as \int_{0}^{1} U_{j}^{(k)} (s, t) d t = {\hat{U}}_{j, \pm}^{(k - 2)} (s) and \int_{- \frac{μ_{j}}{2}}^{\frac{μ_{j}}{2}} U_{j}^{(k)} (s, t) d t = μ_{j} {\hat{U}}_{j, 0}^{(k - 2)} (s) . \end{array}$
Sketch of the proof of Proposition 2 . See the electronic version.
2.3. Matching conditions

2.3.1. Derivation of the matching conditions

To find the missing information (the transmission conditions at the vertices of the graph for the far field terms and the behaviour at infinity for the near field terms), we shall write the so-called matching conditions that ensure that far field and near field expansions coincide in some intermediate areas. Indeed, far field and near field expansions are assumed to be both valid in some intermediate areas $M_{j}^{ε}$ , $j \in Z$ , localized at the left, right and above each junction $K_{j}^{ε}$ , $\begin{matrix} M_{j, m}^{ε} = M_{j, -}^{ε} \cup M_{j, +}^{ε} \cup M_{j, 0}^{ε} . \end{matrix}$

Then the matching zone $M_{j, +}^{ε}$ corresponds to $x - j \to 0$ for the far field and to $X = (x - j) / ε \to + \infty$ for the near field. Typically, we can choose for instance (see Fig. 6 for this example), for any integer j, the left and right intermediate areas $M_{j, -}^{ε}$ and $M_{j, +}^{ε}$ of the form $\begin{matrix} M_{j, -}^{ε} = (0, ε) \times (j - 2 \sqrt{ε}, j - \sqrt{ε}), M_{j, +}^{ε} = (0, ε) \times (j + \sqrt{ε}, j + 2 \sqrt{ε}), \end{matrix}$ and the vertical intermediate areas of the form $\begin{matrix} M_{j, 0}^{ε} = (j - \frac{μ_{j} ε}{2}, j + \frac{μ_{j} ε}{2}) \times (- \frac{L}{2} + \sqrt{ε}, - \frac{L}{2} + 2 \sqrt{ε}) . \end{matrix}$

Fig. 6.

The matching area $M_{j, m}^{ε} = M_{j, -}^{ε} \cup M_{j, +}^{ε} \cup M_{j, 0}^{ε}$ for $j \neq 0$ (NF: near field, FF: far field).

Let us first explain how we proceed in the matching zone $M_{j, +}^{ε}$ which involves the junction $J_{j}$ (for the near field) and the edge $e_{j + \frac{1}{2}}$ (for the far field).

We first use the (formal) Taylor series expansion of the far field term $u_{j + \frac{1}{2}}^{(k)}$ $\begin{matrix} u_{j + \frac{1}{2}}^{(k)} (s) = \sum_{ℓ \in N} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k)} (0) \frac{s^{ℓ}}{ℓ!} \end{matrix}$ so that the ansatz (26) can be (formally) rewritten as $\begin{matrix} (49) & u_{j + \frac{1}{2}}^{ε} (x, y) = \sum_{k \in N} ε^{k} \sum_{ℓ \in N} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k)} (0) \frac{{(x - j)}^{ℓ}}{ℓ!} + O (ε^{\infty}), j \in Z . \end{matrix}$ On the other hand, using the expansion (42) for $U_{j}^{(k)}$ in $B_{j, +}$ , we remark that (31) can be rewritten as $\begin{matrix} (50) & u_{j + \frac{1}{2}}^{ε} (x, y) = \sum_{k \in N} ε^{k} {\hat{U}}_{j, +}^{(k)} (\frac{x - j}{ε}) + O (ε^{\infty}), j \in Z \end{matrix}$ where we have noticed that, for $s = (x - j) / ε$ , $(x - j) \in (0, 1)$ , the terms in factor of $e^{- ℓ π s}$ , $ℓ ⩾ 1$ can be put into the $O (ε^{\infty})$ remainder.

Then, writing $s = (x - j) / ε$ , the identification of (49) and (50) leads to relate the two expansions: $\begin{matrix} \sum_{k \in N} \sum_{ℓ \in N} ε^{k + l} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k)} (0) \frac{s^{ℓ}}{ℓ!} = \sum_{k \in N} ε^{k} {\hat{U}}_{j, +}^{(k)} (s), j \in Z . \end{matrix}$ Finally, identifying the terms of the order $ε^{k}$ leads to the equalities $\begin{matrix} (51) & {\hat{U}}_{j, +}^{(k)} (s) = \sum_{l = 0}^{k} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k - ℓ)} (0) \frac{s^{ℓ}}{ℓ!}, k \in N, j \in Z . \end{matrix}$ Obviously, we proceed in the same way for the matching areas $M_{j, -}^{ε}$ and $M_{j, 0}^{ε}$ to obtain $\begin{matrix} (52) & {\hat{U}}_{j, -}^{(k)} (s) = \sum_{ℓ = 0}^{k} \partial_{s}^{ℓ} u_{j - \frac{1}{2}}^{(k - ℓ)} (1) \frac{s^{ℓ}}{ℓ!}, and {\hat{U}}_{j, 0}^{(k)} (s) = \sum_{ℓ = 0}^{k} \partial_{y}^{ℓ} u_{j}^{(k - ℓ)} (- \frac{L}{2}) \frac{s^{ℓ}}{ℓ!}, k \in N, j \in Z . \end{matrix}$ Using the first equation of (48), the matching conditions (51, 52) are equivalent to $\begin{matrix} (53) & \{\begin{matrix} α_{j, +}^{(k)} = \partial_{s} u_{j + \frac{1}{2}}^{(k - 1)} (0), β_{j, +}^{(k)} = u_{j + \frac{1}{2}}^{(k)} (0), {\hat{U}}_{j, +, part}^{(k - 2)} (s) = \sum_{ℓ = 2}^{k} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k - ℓ)} (0) \frac{s^{ℓ}}{ℓ!}, \\ α_{j, -}^{(k)} = \partial_{s} u_{j - \frac{1}{2}}^{(k - 1)} (1), β_{j, -}^{(k)} = u_{j - \frac{1}{2}}^{(k)} (1), {\hat{U}}_{j, -, part}^{(k - 2)} (s) = \sum_{ℓ = 2}^{k} \partial_{s}^{ℓ} u_{j - \frac{1}{2}}^{(k - ℓ)} (1) \frac{s^{ℓ}}{ℓ!}, \\ α_{j, 0}^{(k)} = \partial_{s} u_{j}^{(k - 1)} (- \frac{L}{2}), β_{j, +}^{(k)} = u_{j}^{(k)} (- \frac{L}{2}), {\hat{U}}_{j, 0, part}^{(k - 2)} (s) = \sum_{ℓ = 0}^{k} \partial_{s}^{ℓ} u_{j}^{(k - ℓ)} (- \frac{L}{2}) \frac{s^{ℓ}}{ℓ!} . \end{matrix} \end{matrix}$

2.3.2. A useful version of the matching conditions

Let us now anticipate here the way we shall exploit these matching conditions for the construction of the asymptotic expansion by induction on k. In fact, these will be used for “building conditions at infinity” for the near field terms in addition to (35). If one assumes $(λ_{m}, U_{m}^{(k)})$ known for $m ⩽ k - 2$ , we know from Proposition 2 and Remark 4: $\begin{matrix} (54) & The functions U_{j, δ, part}^{(k - 2)} (in (43)) are known and considered as data . \end{matrix}$ In view of the decompositions (43)–(44)–(46) of Proposition 2, the equations of the first two columns of (53) can be rewritten as $\begin{matrix} (55) & \{\begin{matrix} U_{j}^{(k)} - (u_{j + \frac{1}{2}}^{(k)} (0) + \partial_{s} u_{j + \frac{1}{2}}^{(k - 1)} (0) s + U_{j, +, part}^{(k - 2)}) \in H arm (B_{j, +}) \cap L_{\exp}^{2} (B_{j, +}), & (i) \\ U_{j}^{(k)} - (u_{j - \frac{1}{2}}^{(k)} (1) + \partial_{s} u_{j - \frac{1}{2}}^{(k - 1)} (1) s + U_{j, -, part}^{(k - 2)}) \in H arm (B_{j, -}) \cap L_{\exp}^{2} (B_{j, -}), & (ii) \\ U_{j}^{(k)} - (u_{j}^{(k)} (- \frac{L}{2}) + \partial_{s} u_{j}^{(k - 1)} (- \frac{L}{2}) s + U_{j, 0, part}^{(k - 2)}) \in H arm (B_{j, 0}) \cap L_{\exp}^{2} (B_{j, 0}), & (iii) \end{matrix} \end{matrix}$ where the spaces $H arm (B_{j, δ})$ are defined in (39) and the spaces $L_{\exp}^{2} (B_{j, δ})$ are given by $\begin{matrix} \{\begin{matrix} L_{\exp}^{2} (B_{j, \pm}) : = {v \in L_{loc}^{2} (B_{j, \pm}), e^{| s |} v \in L^{2} (B_{j, \pm})}, \\ L_{\exp}^{2} (B_{j, 0}) = {v \in L_{loc}^{2} (B_{j, 0}), e^{\frac{| s |}{μ_{j}}} v \in L^{2} (B_{j, 0})} . \end{matrix} \end{matrix}$ Note that the functions belonging to $H arm (B_{j, δ}) \cap L_{\exp}^{2} (B_{j, δ})$ have an expansion of the form (41) with $β_{0, δ} = 0$ . As a result (see the definitions (36)), $\begin{matrix} (56) & H arm (B_{j, δ}) \cap L_{\exp}^{2} (B_{j, δ}) = {v \in H arm (B_{j, δ}), \int_{Σ_{j, δ}} v d σ = 0} . \end{matrix}$ On the other hand, using Remark 5, the condition of the last column of (53) can be reformulated as $\begin{matrix} (57) & \{\begin{matrix} \int_{0}^{1} U_{j, +, part}^{(k - 2)} (s, t) d t = \sum_{ℓ = 2}^{k} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k - ℓ)} (0) \frac{s^{ℓ}}{ℓ!}, & (i) \\ \int_{0}^{1} U_{j, -, part}^{(k - 2)} (s, t) d t = \sum_{ℓ = 2}^{k} \partial_{s}^{ℓ} u_{j - \frac{1}{2}}^{(k - ℓ)} (1) \frac{s^{ℓ}}{ℓ!}, & (ii) \\ \int_{- μ_{j} / 2}^{μ_{j} / 2} U_{j, 0, part}^{(k - 2)} (s, t) d t = μ_{j} \sum_{ℓ = 2}^{k} \partial_{s}^{ℓ} u_{j}^{(k - ℓ)} (- \frac{L}{2}) \frac{s^{ℓ}}{ℓ!} . & (iii) \end{matrix} \end{matrix}$ However, these conditions are redundant, as explained in the following lemma, in the sense that at each step k, they appear to be consequences of the matching conditions (55) for $m ⩽ k - 1$ .

Lemma 2.
Assume that ( 35 )–( 30 ) are satisfied (with m instead of k) for $m ⩽ k$ and that the matching conditions ( 53 ) (with m instead of k) are satisfied for $m ⩽ k - 1$ . Then, ( 57 ) is satisfied.
Proof.
See the electronic version. □

In other words, under the hypotheses of the previous lemma, it suffices to satisfy (55) in order to satisfy the matching conditions (53).
3. Analysis of far and near field equations

As usual in the method of asymptotic expansions, the construction of the terms ${λ^{(k)}, u^{(k)}, U^{(k)}}$ in the formal expansions will be done by induction on k. The idea is, at each step of the induction process to “eliminate” the near field $U^{(k)}$ in order to formulate a problem in ${λ^{(k)}, u^{(k)}}$ only, which is similar in its form to the problem (2.2) for ${λ^{(0)}, u^{(0)}}$ . To do so, in Section 3.1, we first derive transmission conditions on the far fields terms ${u^{(k)}}$ assuming that the whole sequence ${λ^{(k)}, u^{(k)}, U^{(k)}}$ exists (Prop. 3). The next two sections prepare the induction process which will be described in Section 4. In Section 3.2, we prove that the above mentioned transmission conditions are also sufficient conditions for the construction of $U^{(k)}$ , assuming that the previous terms ${λ^{(m)}, u^{(m)}, U^{(m)}}$ , $m < k$ are known as well as the far field term $u^{(k)}$ (Prop. 4). Finally, in Section 3.3, we formulate the problem in $(λ^{(k)}, u^{(k)})$ and show the existence and uniqueness of the solution provided that ${λ^{(m)}, u^{(m)}, U^{(m)}}$ , $m < k$ are known (Prop. 5). The step-by step construction of each term of the asymptotic expansion is conducted in Section 4.

3.1. Necessary conditions for the existence of the near field terms

In this section, assuming the existence of the whole sequence ${λ^{(k)}, u^{(k)}, U^{(k)}}_{k \in N}$ , we derive non homogeneous transmission conditions for $u^{(k)}$ (similar to those necessarily satisfied by for $u^{(0)}$ ). To obtain these conditions, several approaches are possible. We have chosen the one that consists in “replacing” the problem satisfied by each near field term (namely (35) plus the matching conditions (55)(57)) by an “equivalent” problem set in the bounded domain $K_{j}$ defined in (32) using the so called Dirichlet-to-Neumann operators introduced in the next section. Beyond pure theoretical purposes, another interest of this method is that it directly results into a numerical method that will be explained later.

3.1.1. Dirichlet to Neumann operators

Let $T_{j}^{δ} \in L (H^{\frac{1}{2}} (Σ_{j, δ}), H^{- \frac{1}{2}} (Σ_{j, δ}))$ be the DtN operator defined by $\begin{matrix} (58) & T_{j}^{\pm} φ = - \partial_{n} u_{j}^{\pm} (φ) on Σ_{j, \pm}, \end{matrix}$ where $u_{j}^{\pm} (φ) \in H^{1} (B_{j, δ})$ is the unique solution of (37) and n denotes the unit normal vector to $Σ_{j, δ}$ outgoing with respect to $K_{j}$ . Formula (41) allows us to put the operators $T_{j}^{δ}$ in diagonal form $\begin{matrix} (59) & T_{j}^{\pm} φ = \sum_{ℓ = 1}^{+ \infty} ℓ π {(φ, v_{ℓ})}_{L^{2} (Σ_{j, \pm})} v_{ℓ}, T_{j}^{0} φ = \sum_{ℓ = 1}^{+ \infty} \frac{ℓ π}{μ_{j}} {(φ, v_{ℓ, 0}^{j})}_{L^{2} (Σ_{j, 0})} v_{ℓ, 0}^{j} . \end{matrix}$ Note that, by definition of the spaces $H arm (B_{δ, j})$ and the operators $T_{j}^{δ}$ (and some abuse of notation), $\begin{matrix} (60) & \forall U \in H arm (B_{δ, j}), \partial_{n} U + T_{j}^{δ} U = 0 on Σ_{δ, j} . \end{matrix}$

3.1.2. Reduction to a bounded domain

We wish to characterize, for $j \in Z$ and $k \in N$ , the restriction of $U_{j}^{(k)}$ to $K_{j}$ , denoted ${\overset{ˇ}{U}}_{j}^{(k)}$ for simplicity. $\begin{matrix} (61) & We first introduce the notation: Φ_{j}^{(k - 2)} : = - \sum_{m = 0}^{k - 2} λ^{(k - m - 2)} U_{j}^{(m)} \end{matrix}$ where again the superscript $(k - 2)$ is here to indicate that this function depends on the $(λ_{m}, U_{m})$ ’s for $m ⩽ k - 2$ and is thus a data. Of course, we immediately infer from (35) that $\begin{matrix} (62) & \{\begin{matrix} Δ {\overset{ˇ}{U}}_{j}^{(k)} = Φ_{j}^{(k - 2)} in K_{j}, \\ \partial_{n} {\overset{ˇ}{U}}_{j}^{(k)} = 0 on \partial K_{j} ∖ (Σ_{j, +} \cup Σ_{j, -} \cup Σ_{j, 0}) . \end{matrix} \end{matrix}$ To close the problem, it remains to derive boundary conditions on the boundaries $Σ_{j, δ}$ , $δ \in {0, +, -}$ . More precisely, we write non homogeneous DTN conditions on each boundary $Σ_{j, δ}$ . Noticing that constant functions belong to $H arm (B_{δ, j})$ , we deduce from (55) and the property (60) that $\begin{array}{l} (63) & \partial_{n} U_{j}^{(k)} + T_{j}^{δ} U_{j}^{(k)} = g_{j, δ}^{(k - 1)} on Σ_{j, δ}, where \\ (64) & \{\begin{matrix} g_{j, +}^{(k - 1)} = \partial_{s} u_{j + \frac{1}{2}}^{(k - 1)} (0) + g_{j, +, part}^{(k - 2)}, g_{j, +, part}^{(k - 2)} = (\partial_{n} + T_{j}^{+}) U_{j, +, part}^{(k - 2)} on Σ_{j, +}, \\ g_{j, -}^{(k - 1)} = - \partial_{s} u_{j - \frac{1}{2}}^{(k - 1)} (1) + g_{j, -, part}^{(k - 2)}, g_{j, -, part}^{(k - 2)} = (\partial_{n} + T_{j}^{-}) U_{j, -, part}^{(k - 2)} on Σ_{j, -}, \\ g_{j, 0}^{(k - 1)} = \partial_{y} u_{j}^{(k - 1)} (- \frac{L}{2}) + g_{j, 0, part}^{(k - 2)}, g_{j, 0, part}^{(k - 2)} = (\partial_{n} + T_{j}^{0}) U_{j, 0, part}^{(k - 2)} on Σ_{j, 0} . \end{matrix} \end{array}$ Collecting (62) and (63) we see that, in $K_{j}$ , ${\overset{ˇ}{U}}_{j}^{(k)}$ satisfies the boundary value problem $\begin{matrix} (65) & \{\begin{matrix} Δ {\overset{ˇ}{U}}_{j}^{(k)} = Φ_{j}^{(k - 2)} & in K_{j}, \\ \partial_{n} {\overset{ˇ}{U}}_{j}^{(k)} + T_{j}^{δ} {\overset{ˇ}{U}}_{j}^{(k)} = g_{j, δ}^{(k - 1)} & on Σ_{j, δ}, δ = 0, +, - \\ \partial_{n} {\overset{ˇ}{U}}_{j}^{(k)} = 0 & on \partial K_{j} ∖ (Σ_{j, +} \cup Σ_{j, -} \cup Σ_{j, 0}) . \end{matrix} \end{matrix}$ Note that, since the image of constant functions by any of the operators $T_{j}^{δ}$ is 0, any constant is a solution of the homogeneous boundary value problem corresponding to (65), which means that, at best, a solution of (65) is defined up to an additive constant. As a matter of fact, the study of (65) relies on Fredholm’s alternative (exactly as the Neumann Laplace problem) and we let the reader prove the following result:

Lemma 3.
For $j \in Z$ , let $Φ \in L^{2} (K_{j})$ , $g_{δ} \in H^{- 1 / 2} (Σ_{j, δ})$ . There exists a unique solution $U \in H^{1} (K_{j}) / R$ to the problem $\begin{matrix} (66) & \{\begin{matrix} Δ U = Φ in K_{j}, \\ \partial_{n} U + T_{j}^{δ} U = g_{δ} on Σ_{j, δ}, δ = 0, +, -, \\ \partial_{n} U = 0 on \partial K_{j} ∖ (Σ_{j, +} \cup Σ_{j, -} \cup Σ_{j, 0}) \end{matrix} \end{matrix}$ if and only if the following compatibility condition is satisfied, ${⟨ \cdot, \cdot ⟩}_{Σ_{j, δ}}$ denoting the duality bracket between $H^{- 1 / 2} (Σ_{j, δ})$ and $H^{1 / 2} (Σ_{j, δ})$ . $\begin{matrix} (67) & {⟨ g_{+}, 1 ⟩}_{Σ_{j, +}} + {⟨ g_{-}, 1 ⟩}_{Σ_{j, -}} + {⟨ g_{0}, 1 ⟩}_{Σ_{j, 0}} = \int_{K_{j}} Φ . \end{matrix}$

The function U solution of (66) is omny defined up to an additive constant at our disposal.

As we shall see later, in the case of $U_{j}^{(k)}$ , the choice of this constant will imposed by the matching conditions.
3.1.3. Necessary conditions on the far field

The main result of this section is Proposition 3 whose proof involves two “profile functions” $N_{j}^{\pm}$ :

The function $N_{j}^{-}$ is the unique function of $H^{1} (K_{j})$ solution to the problem (66) with $\begin{matrix} (68) & Φ = 0, g_{+} = 0, g_{-} = 1, g_{0} = - 1 / μ_{j} that satisfies \int_{K_{j}} N_{j}^{-} d x = 0 . \end{matrix}$

The function $N_{j}^{+}$ is the unique function of $H^{1} (K_{j})$ solution to the problem (66) with $\begin{matrix} (69) & Φ = 0, g_{+} = - 1, g_{-} = 0, g_{0} = 1 / μ_{j} that satisfies \int_{K_{j}} N_{j}^{+} d x = 0 . \end{matrix}$

Remark 6.
Note that, according to the identification of the junctions $J_{j}$ , $j \neq 0$ to the single junction $J_{1}$ , all profile functions $N_{j}^{\pm}$ coincide with $N_{1}^{\pm}$ . As a consequence, we only have four profile functions for $j = 0, 1$ and ±. In addition, $N_{j}^{+}$ and $N_{j}^{-}$ are linked by the following symmetry property: $\begin{matrix} (70) & N_{j}^{-} (X, Y) = N_{j}^{+} (- X, Y) \forall j \in N . \end{matrix}$
Proposition 3.
Assume the existence of a sequence ${λ^{(k)}, u^{(k)}, U^{(k)}}$ satisfying ( 30 ), ( 35 ), ( 55 ) and ( 57 ). Then, the far field $u^{(k)}$ satisfies for any $k \in N$ the inhomogeneous generalized Kirchhoff conditions $\begin{matrix} (71) & \partial_{s} u_{j + \frac{1}{2}}^{(k)} (0) - \partial_{s} u_{j - \frac{1}{2}}^{(k)} (1) + μ_{j} \partial_{y} u_{j}^{(k)} (- \frac{L}{2}) = Ξ_{j}^{(k - 1)}, \end{matrix}$ as well as the inhomogeneous jump conditions $\begin{matrix} (72) & \begin{matrix} u_{j - \frac{1}{2}}^{(k)} (1) - u_{j}^{(k)} (- \frac{L}{2}) = Δ_{j, -}^{(k - 1)}, j \in Z, & (i) \\ u_{j}^{(k)} (- \frac{L}{2}) - u_{j + \frac{1}{2}}^{(k)} (0) = Δ_{j, +}^{(k - 1)}, j \in Z, & (ii) \end{matrix} \end{matrix}$ where, $Φ_{j}^{(k)}$ , $g_{j, δ}^{(k)}$ and $g_{j, δ, part}^{(k)}$ being given by ( 61 , 64 ), $\begin{matrix} (73) & \{\begin{matrix} Ξ_{j}^{(k - 1)} = - {⟨ g_{j, +, part}^{(k - 1)}, 1 ⟩}_{Σ_{j, +}} - {⟨ g_{j, -, part}^{(k - 1)}, 1 ⟩}_{Σ_{j, -}} - {⟨ g_{j, 0, part}^{(k - 1)}, 1 ⟩}_{Σ_{j, 0}} + \int_{K_{j}} Φ_{j}^{(k - 1)}, & (i) \\ \begin{matrix} Δ_{j, -}^{(k - 1)} & = \sum_{ℓ = 1}^{k} (\frac{1}{ℓ!} \partial_{y}^{ℓ} u_{j}^{(k - ℓ)} (- \frac{L}{2}) - \frac{{(- 1)}^{ℓ} {μ_{j}}^{ℓ}}{2^{ℓ} ℓ!} \partial_{s}^{ℓ} u_{j - \frac{1}{2}}^{(k - ℓ)} (1)) \\ + \sum_{δ \in {+, -, 0}} {⟨ g_{j, δ}^{(k - 1)}, N_{j}^{-} ⟩}_{Σ_{j, δ}} - \int_{K_{j}} Φ_{j}^{(k - 2)} N_{j}^{-}, \end{matrix} & (ii) \\ \begin{matrix} Δ_{j, +}^{(k - 1)} & = \sum_{ℓ = 1}^{k} (\frac{{μ_{j}}^{ℓ}}{2^{ℓ} ℓ!} \partial_{s}^{ℓ} u_{j + \frac{1}{2}}^{(k - ℓ)} (0) - \frac{1}{ℓ!} \partial_{y}^{ℓ} u_{j}^{(k - ℓ)} (- \frac{L}{2})) \\ + \sum_{δ \in {+, -, 0}} {⟨ g_{j, δ}^{(k - 1)}, N_{j}^{+} ⟩}_{Σ_{j, δ}} - \int_{K_{j}} Φ_{j}^{(k - 2)} N_{j}^{+} . \end{matrix} & (iii) \end{matrix} \end{matrix}$
Proof.
See the electronic version. □

3.2. Towards the inductive construction of the near field terms

In this section, we assume that we are inside an induction process and wish to construct $U^{(k)}$ for $k ⩾ 1$ assuming that $\begin{matrix} (H_{k}) \{\begin{matrix} The far field terms (u^{(m)}, λ^{(m)}), m ⩽ k and the near field terms U^{(m)}, m ⩽ k - 1 \\ have been constructed so as to satisfy (30), (71) and (72) on the one hand, (35), (53) \\ on the other hand. \end{matrix} \end{matrix}$

Proposition 4.
Assume that $(H_{k})$ holds. Then, for each j, there exists a unique near field term $U_{j}^{(k)} \in V_{j}$ satisfying ( 35 ) and ( 55 ).
Proof.
See the electronic version. □

3.3. The problem in $(u^{(k)}, λ^{(k)})$

Here we assume that ${λ^{(m)}, u^{(m)}, U^{(m)}}$ are known for $m < k$ . Collecting the results of the previous sections (far field equations (30), Kirchhoff conditions (71), and non-homogeneous jump conditions (72)), we see that $(u^{(k)}, λ^{(k)})$ should satisfy the following problem: $\begin{matrix} (74) & \forall j \in Z, \{\begin{matrix} \partial_{s}^{2} u_{j + \frac{1}{2}}^{(k)} (s) + λ^{(0)} u_{j + \frac{1}{2}}^{(k)} (s) = - λ^{(k)} u_{j + \frac{1}{2}}^{(0)} (s) - f_{j + \frac{1}{2}}^{(k - 1)} (s), s \in (0, 1), & (i) \\ \partial_{y}^{2} u_{j}^{(k)} (y) + λ^{(0)} u_{j}^{(k)} (y) = - λ^{(k)} u_{j}^{(0)} (y) - f_{j}^{(k - 1)} (y), y \in (- L / 2, 0), & (ii) \\ \partial_{y} u_{j}^{(k)} (0) = 0, & (iii) \\ \partial_{s} u_{j + \frac{1}{2}}^{(k)} (0) - \partial_{s} u_{j - \frac{1}{2}}^{(k)} (1) + μ_{j} \partial_{y} u_{j}^{(k)} (- \frac{L}{2}) = Ξ_{j}^{(k - 1)}, & (iv) \\ u_{j - \frac{1}{2}}^{(k)} (1) - u_{j}^{(k)} (- \frac{L}{2}) = Δ_{j, -}^{(k - 1)}, u_{j}^{(k)} (- \frac{L}{2}) - u_{j + \frac{1}{2}}^{(k)} (0) = Δ_{j, +}^{(k - 1)} & (v) \end{matrix} \end{matrix}$ where the data at the right hand sides of last two equations are given by (73) and (64), and thus known from ${λ^{(m)}, u^{(m)}, U^{(m)}}$ , $m < k$ , while $\begin{array}{l} (75) & f_{j + \frac{1}{2}}^{(k - 1)} : = \sum_{m = 1}^{k - 1} λ^{(k - m)} u_{j + \frac{1}{2}}^{(m)}, f_{j}^{(k - 1)} : = \sum_{m = 1}^{k - 1} λ^{(k - m)} u_{j}^{(m)}, j \in Z . \end{array}$ Let $E (λ_{0}) : = span [u^{(0)}] \subset H^{1} (G^{-}) \subset H_{br}^{1} (G^{-})$ be the eigenspace of $A_{s}^{μ}$ associated to the eigenvalue $λ_{0}$ .

Proposition 5.
Let $k ⩾ 1$ . Assume that $\begin{matrix} (76) & f^{(k - 1)} \in L_{2}^{μ} (G^{-}), {Ξ_{j}^{(k - 1)}}_{j \in Z} \in l_{2} (Z) and {Δ_{j, \pm}^{(k - 1)}}_{j \in Z} \in l_{2} (Z), \end{matrix}$ then there exists a unique $(u^{(k)}, λ^{(k)}) \in H_{br}^{1} (G^{-}) / E (λ_{0}) \times R$ solution of ( 74 ). Moreover $\begin{array}{l} (77) & λ^{(k)} = {‖ u^{(0)} ‖}_{L_{2}^{μ} (G^{-})}^{- 2} (2 \sum_{j \in Z} {\tilde{Ξ}}_{j}^{(k - 1)} u_{j}^{(0)} - {(f^{(k - 1)}, u^{(0)})}_{L_{2}^{μ} (G^{-})}), \\ (78) & where {\tilde{Ξ}}_{j}^{(k - 1)} = Ξ_{j}^{(k - 1)} - \frac{\sqrt{λ^{(0)}}}{sin \sqrt{λ^{(0)}}} (Δ_{j + 1, -}^{(k - 1)} - Δ_{j - 1, +}^{(k - 1)} + cos \sqrt{λ^{(0)}} (Δ_{j, +}^{(k - 1)} - Δ_{j, -}^{(k - 1)})) . \end{array}$
Proof.
The proof of this theorem is an elementary application of the Fredholm’s alternative. A similar proof can be found, for instance in [20] (see Theorem 4.10, Corollary 2.2 and Theorem 2.13). Note that (77) is the necessary compatibility condition for the existence of a solution of (74) seen as a generalized boundary value problem for $u^{(k)}$ . □
Remark 7.
In the solution of (74), the field $u^{(k)}$ is defined up to the addition of any element of $E (λ_{0})$ . The uniqueness can be restored by imposing, for instance, any linear condition of the form $ℓ (u^{k}) = 0$ , where ℓ is a linear form of $H_{br}^{1} (G^{-})$ such that $ℓ (u^{0}) \neq 0$ . The choice of ℓ will specify the construction of a particular pseudo-mode. In Section 4 (see e.g. (104)), we shall precise our choice of ℓ.

Besides, we can notice that the expression of $λ^{(k)}$ given by (77) is homogeneous of degree 0 with respect to the eigenvector $u^{(0)}$ so that one can check that it does not depend on its normalization. Indeed, it can be shown by induction that $f_{j}^{(k - 1)}$ and ${\tilde{Ξ}}_{j}^{(k - 1)}$ depend linearly on $u^{(0)}$ .

We conclude this section by a symmetry property (in the variable x) of the far field $u^{(k)}$ . Lemma 4.
Let $k \in N^{}$ . Assume that (* 76 ) holds and that condition ( 77 ) is fulfilled. Suppose also that the following symmetry conditions hold: $\forall j \in Z$ , $\begin{matrix} (79) & (i) \{\begin{matrix} f_{- j - \frac{1}{2}}^{(k - 1)} (1 - s) = f_{j + \frac{1}{2}}^{(k - 1)} (s) s \in [0, 1] \\ f_{- j}^{(k - 1)} (y) = f_{j}^{(k - 1)} (y) y \in [- \frac{L}{2}, 0] \end{matrix} (ii) \{\begin{matrix} Δ_{j, +}^{(k - 1)} = - Δ_{- j, -}^{(k - 1)} \\ Ξ_{j}^{(k - 1)} = Ξ_{- j}^{(k - 1)} . \end{matrix} \end{matrix}$ Then, the solution $u^{(k)}$ of Problem ( 74 ) is symmetric, i.e., for any $j \in Z$ , $\begin{matrix} (80) & \{\begin{matrix} u_{- j - \frac{1}{2}}^{(k)} (1 - s) = u_{j + \frac{1}{2}}^{(k)} (s) s \in [0, 1] \\ u_{- j}^{(k)} (y) = u_{j}^{(k)} (y) y \in [- \frac{L}{2}, 0] \end{matrix} \end{matrix}$
Proof.
See the electronic version. □

4. The asymptotic expansion: Existence and algorithm

By repeating applications of Proposition 5 and Proposition 4 (successively), we are able to define a recursive procedure to construct all the terms of the different asymptotic expansions (far field expansion, near field expansion and eigenvalue expansion) up to any order. The construction is done by induction.

Moreover, we can derive explicit formulas for the far field terms and semi-explicit expressions for the near field terms, which are suitable for the numerical computations of the successive terms of the asymptotic expansion. In particular, we point out two important features of the forthcoming construction:

First, by induction, all far field terms $(u_{j + 1 / 2}^{(k)}, u_{j}^{(k)})$ inherit of the symmetry property (80). Moreover, the near field term satisfy an analogous symmetry property $\begin{matrix} (81) & U_{- j}^{(k)} (X, Y) = U_{j}^{(k)} (- X, Y) \forall j \in N . \end{matrix}$ In practice, at each step k, it is consequently sufficient to compute $u_{j + 1 / 2}^{(k)}$ , $u_{j}^{(k)}$ and $U_{j}^{(k)}$ for $j \in N$ .

Secondly, an explicit dependance with respect to j can be proved by induction. This turns out to be very useful from the numerical point of view.

This section is organized as follows. First, we initialize the induction process for

k = 0

in Section 4.1. Then, we proceed to the first induction step

k = 1

in Section 4.2. This step has a pedagogical interest for the understanding of the general induction step at any order k made in Section 4.3.

4.1. Order 0: Initialization of the algorithm

We start from an eigenvalue $λ^{(0)}$ of the operator $A_{s}^{μ}$ defined in the limit graph and the associated symmetric eigenvector $u^{(0)}$ given by (23–24).

For convenience in the forthcoming exposition, we shall use the following alternative expressions of $(u_{j + \frac{1}{2}}^{(0)}, u_{j}^{(0)})$ for $j ⩾ 0$ , (completed by the symmetry property (80)) $\begin{array}{l} (82) & \{\begin{matrix} u_{j + \frac{1}{2}}^{(0)} (s) = r^{j} (a_{0}^{(0)} cos (\sqrt{λ^{(0)}} s) + b_{0}^{(0)} sin (\sqrt{λ^{(0)}} s)), s = x - j \in [0, 1], \\ u_{j}^{(0)} (y) = r^{j} c_{0}^{(0)} cos (\sqrt{λ^{(0)}} y), y \in [- L / 2, 0], \end{matrix} \\ where a_{0}^{(0)} = 1, b_{0}^{(0)} = sin {(\sqrt{λ^{(0)}})}^{- 1} {((r - cos (\sqrt{λ^{(0)}})), c_{0}^{(0)} = cos (\sqrt{λ^{(0)}} L / 2))}^{- 1} . \end{array}$ According to Section 3 and more particularly Proposition 4, each near field terms $U_{j}^{(0)}$ , $j \in Z$ is then defined as the unique solution of (35, 55) for $k = 0$ . Indeed, thanks to the convention of Remark 3, it is easy to check that Assumption $(H_{0})$ (needed for applying Proposition 4) is nothing but the fact that $u^{(0)}$ is an eigenvector associated to the eigenvalue $λ^{(0)}$ , which is precisely our starting point.

Moreover, since $Φ^{(- 1)} = 0$ , it is then easy to see that for all j, $U_{j}^{(0)}$ is constant equal to $u_{j + \frac{1}{2}}^{(0)} (0)$ , i.e. $\begin{matrix} (83) & U_{j}^{(0)} = r^{| j |}, \forall j \in Z . \end{matrix}$

4.2. Order 1: First induction step

We shall construct in turn

the coefficient $λ^{(1)}$ and the far field terms $u^{(1)}$ by means of an explicit resolution (see Remark 8) of (74) for $k = 1$ . We prove that $u^{(1)}$ is symmetric (in the sense of (80)) and that $\begin{matrix} (84) & \{\begin{matrix} u_{j + \frac{1}{2}}^{(1)} (s) = r^{j} \sum_{ℓ = 0}^{1} s^{ℓ} (a_{ℓ}^{(1)} (j) cos (\sqrt{λ_{0}} s) + b_{ℓ}^{(1)} (j) sin (\sqrt{λ_{0}} s)), s \in [0, 1], j \in N, \\ u_{j}^{(1)} (y) = r^{j} \sum_{ℓ = 0}^{1} y^{ℓ} (c_{ℓ}^{(1)} (j) cos (\sqrt{λ_{0}} y) + d_{ℓ}^{(1)} (j) sin (\sqrt{λ_{0}} y)), y \in [- \frac{L}{2}, 0], j \in N^{*}, \\ u_{0}^{(1)} (y) = \sum_{ℓ = 0}^{1} y^{ℓ} (c_{ℓ, 0}^{(1)} cos (\sqrt{λ_{0}} y) + d_{ℓ, 0}^{(1)} sin (\sqrt{λ_{0}} y)), y \in [- \frac{L}{2}, 0] \end{matrix} \end{matrix}$ where the coefficients $(a_{ℓ}^{(1)} (j), d_{ℓ}^{(1)} (j), c_{ℓ}^{(1)} (j), d_{ℓ}^{(1)} (j), c_{ℓ, 0}^{(1)}, d_{ℓ, 0}^{(1)})$ are explicitly determined.

the near field term $U^{(1)}$ , which is symmetric in the sense of (81) and is of the form $\begin{matrix} (85) & U_{j}^{(1)} (\cdot) = r^{j} (U^{(1)} (\cdot) + P^{(1)} (j)) \forall j \in N^{*}, with J_{j} identified to J_{1} \end{matrix}$ where $U^{(1)} \in H_{loc}^{1} (J_{1})$ is a so-called profile function and the constant $P^{(1)} (j)$ is a polynomial of degree 1 in j. Note that (85) avoids the case $j = 0$ which will be the object of a separate treatment.

Remark 8.
To avoid a boring exposition of long and complicated formulas or expressions, we shall most often restrict ourselves to explain how these explicit computations can be done, without giving the results (this will be also the case in Section 4.3). Note however that these formulas are necessary and used in the numerical method presented in Section 6, while the general form of these formulas will be used for the error analysis of Section 5.
Remark 9.
Note that it is natural that the vertical edge corresponding to $j = 0$ is treated in a separate manner since it corresponds to the refined branch of the thich graph $Ω_{ε}^{μ, -}$ .

4.2.1. Determination of $λ^{(1)}$ and $u^{(1)}$

They are obtained by solving (74) for $k = 1$ . Let us investigate the structure of the data $\begin{matrix} (f_{j + 1 / 2}^{(0)}, f_{j}^{(0)}, Δ_{j, +}^{(0)}, Δ_{j, -}^{(0)}, Ξ_{j}^{(0)}) \end{matrix}$ of this problem. First of all, by definition, we know that $\begin{matrix} (86) & \forall j \in Z, f_{j + 1 / 2}^{(0)} = f_{j}^{(0)} = 0 . \end{matrix}$ Concerning $Δ_{j, \pm}^{(0)}$ , we first notice that, from (64), (82) and Remarks 3 and 6, $\begin{matrix} (87) & \forall j ⩾ 0, g_{j, δ}^{(0)} = r^{j} g_{δ}^{(0)}, and \forall j \in Z, g_{- j, 0}^{(0)} = g_{j, 0}^{(0)} g_{- j, +}^{(0)} = g_{j, -}^{(0)} \end{matrix}$ where moreover $\begin{matrix} (88) & g_{+}^{(0)} = - g_{-}^{(0)} = b_{0}^{(0)} \sqrt{λ^{(0)}}, g_{0}^{(0)} = c_{0}^{(0)} \sqrt{λ^{(0)}} sin (\sqrt{λ^{(0)}} L / 2) . \end{matrix}$ Using the definition (73)(ii) and (iii) for $Δ_{j, \pm}^{(0)}$ , we easily see, since $Φ^{(- 1)} = 0$ , that $\begin{matrix} \{\begin{matrix} Δ_{j, -}^{(0)} = \partial_{s} u_{j}^{(0)} (- \frac{L}{2}) + \frac{μ_{j}}{2} \partial_{s} u_{j - \frac{1}{2}}^{(0)} + \sum_{δ \in {+, -, 0}} {⟨ g_{j, δ}^{(0)}, N_{j}^{-} ⟩}_{Σ_{j, δ}}, \\ Δ_{j, +}^{(0)} = \frac{μ_{j}}{2} \partial_{s} u_{j + \frac{1}{2}}^{(0)} - \partial_{s} u_{j}^{(0)} (- \frac{L}{2}) + \sum_{δ \in {+, -, 0}} {⟨ g_{j, δ}^{(0)}, N_{j}^{+} ⟩}_{Σ_{j, δ}} . \end{matrix} \end{matrix}$ Therefore, as a consequence of (82) and (87)–(88), we can write $\begin{matrix} (89) & Δ_{j, \pm}^{(0)} = r^{j} {\hat{Δ}}_{\pm}^{(0)} j \in N \end{matrix}$ where ${\hat{Δ}}_{\pm}^{(0)}$ can be explicitly determined as a function of $(a_{0}^{(0)}, b_{0}^{(0)}, c_{0}^{(0)})$ and $λ^{(0)}$ . Moreover, using the symmetry properties of $N_{j}^{\pm}$ (see Remark 6), we can see that for $j \in Z$ , $\begin{matrix} (90) & Δ_{- j, -}^{(0)} = - Δ_{j, +}^{(0)} and Δ_{- j, +}^{(0)} = - Δ_{j, -}^{(0)} . \end{matrix}$ In particular we deduce that ${Δ_{j, \pm}^{(0)}}_{j \in Z} \in l_{2} (Z)$ . Finally, we prove below that $\begin{matrix} (91) & \forall j \in Z, Ξ_{j}^{(0)} = 0 . \end{matrix}$ Indeed, since $Φ_{j}^{(0)} = - λ^{(0)} U_{j}^{(0)}$ (see (61)) and $U_{j}^{(0)} \equiv r^{| j |}$ (the constant function), using definition (73)(i), we obtain (we use also $meas K_{j} = μ_{j}$ ) $\begin{matrix} (92) & Ξ_{j}^{(0)} = - {⟨ g_{j, +, part}^{(0)}, 1 ⟩}_{Σ_{j, +}} - {⟨ g_{j, -, part}^{(0)}, 1 ⟩}_{Σ_{j, -}} - {⟨ g_{j, 0, part}^{(0)}, 1 ⟩}_{Σ_{j, 0}} - μ_{j} λ^{(0)} r^{| j |} . \end{matrix}$ Using the definition (64) for $g_{j, 0, part}^{(0)}$ and the definition of $T_{j}^{0}$ , we have $\begin{matrix} {⟨ g_{j, 0, part}^{(0)}, 1 ⟩}_{Σ_{j, 0}} = {⟨ (\partial_{n} + T_{j}^{0}) U_{j, 0, part}^{(0)}, 1 ⟩}_{Σ_{j, 0}} \equiv {⟨ \partial_{n} U_{j, 0, part}^{(0)}, 1 ⟩}_{Σ_{j, 0}} . \end{matrix}$ Next, according to Proposition 2, $U_{j, 0, part}^{(0)}$ is obtained from the decomposition (43) of $U_{j}^{(2)}$ for $k = 2$ and is solution of (45)–(46)–(47). It is then easy to see that $\begin{matrix} U_{j, 0, part}^{(0)} (s, t) = - λ^{(0)} r^{| j |} \frac{s^{2}}{2} + \sum_{ℓ = 1}^{+ \infty} {\hat{U}}_{j, ℓ, 0, part}^{(0)} (s) e^{- \frac{ℓ π}{μ_{j}} | s |} v_{ℓ, 0}^{j} (t) . \end{matrix}$ Then, since $Σ_{j, 0}$ corresponds to $s = 1$ , we find that $\begin{matrix} (93) & {⟨ g_{j, 0, part}^{(0)}, 1 ⟩}_{Σ_{j, 0}} = - λ^{(0)} r^{| j |} μ_{j} . \end{matrix}$ Proceeding in the same way in the bands $B_{j, \pm}$ where $\partial_{n} = \pm \partial_{s}$ , we find that $\begin{matrix} (94) & {⟨ g_{j, +, part}^{(0)}, 1 ⟩}_{Σ_{j, +}} = - \frac{μ_{j}}{2} λ^{(0)} r^{| j |}, {⟨ g_{j, -, part}^{(0)}, 1 ⟩}_{Σ_{j, -}} = \frac{μ_{j}}{2} λ^{(0)} r^{| j |} . \end{matrix}$ The relations (93) and (94) enables to conclude that $Ξ_{j}^{(0)} = 0$ for all $j \in Z$ .

Thanks to (86), (89) and (91), the assumptions of Proposition 5 are satisfied for $k = 1$ so that we can claim that there exists a unique $(λ^{(1)}, u^{(1)})$ , up to a normalization condition for $u^{(1)}$ (cf. Remark 7) that we will specify below (see (104)). Moreover, according to (77), $λ^{(1)}$ is given by $\begin{matrix} (95) & λ^{(1)} = 2 {‖ u^{(0)} ‖}_{L_{2}^{μ} (G^{-})}^{- 2} (\sum_{j \in Z} {\tilde{Ξ}}_{j}^{(0)} u_{j}^{(0)}), \end{matrix}$ where, according to (78) and (89), we compute that $\begin{matrix} \forall j \in N, {\tilde{Ξ}}_{j}^{(0)} = - r^{j} \frac{\sqrt{λ^{(0)}}}{sin \sqrt{λ^{(0)}}} (r {\hat{Δ}}_{-}^{(0)} - r^{- 1} {\hat{Δ}}_{+}^{(0)} + cos \sqrt{λ^{(0)}} ({\hat{Δ}}_{+}^{(0)} - {\hat{Δ}}_{-}^{(0)})) . \end{matrix}$ We can also give an explicit formula for $u^{(1)}$ . First, from properties (90), we can claim that $u^{(1)}$ satisfy the symmetry property (80), so that we can restrict ourselves in the following to $j ⩾ 0$ .

Let us first consider the two linear ordinary differential equations of (74) for $k = 1$ , namely: $\begin{matrix} (96) & \{\begin{matrix} \partial_{s}^{2} u_{j + \frac{1}{2}}^{(1)} (s) + λ^{(0)} u_{j + \frac{1}{2}}^{(1)} (s) = - λ^{(1)} u_{j + \frac{1}{2}}^{(0)} (s), s \in (0, 1), \\ \partial_{y}^{2} u_{j}^{(1)} (y) + λ^{(0)} u_{j}^{(1)} (y) = - λ^{(1)} u_{j}^{(0)} (y), y \in (- L / 2, 0), \end{matrix} j \in N . \end{matrix}$ Using (82), we first compute a particular solution of (96), namely $\begin{matrix} (97) & \{\begin{matrix} u_{j + \frac{1}{2}, part}^{(1)} (s) = λ^{(1)} r^{j} \frac{1}{2 \sqrt{λ^{(0)}}} (- a_{0}^{(0)} s sin (\sqrt{λ^{(0)}} s) + b_{0}^{(0)} s cos (\sqrt{λ^{(0)}} s)), \\ u_{j, part}^{(1)} (y) = λ^{(1)} r^{j} \frac{1}{2 \sqrt{λ^{(0)}}} (- c_{0}^{(0)} y sin (\sqrt{λ^{(0)}} y)), \end{matrix} j \in N . \end{matrix}$ Therefore, there exist ${{\overset{ˇ}{a}}_{0}^{(1)} (j), {\overset{ˇ}{b}}_{0}^{(1)} (j), j \in N}, {{\overset{ˇ}{c}}_{0}^{(1)} (j), {\overset{ˇ}{d}}_{0}^{(1)} (j), j > 0}$ and ${{\overset{ˇ}{c}}_{0, 0}^{(1)}, {\overset{ˇ}{d}}_{0, 0}^{(1)}}$ such that $\begin{array}{rcl} (98) & \{\begin{matrix} u_{j + \frac{1}{2}}^{(1)} (s) = {\overset{ˇ}{a}}_{0}^{(1)} (j) cos (\sqrt{λ^{(0)}} s) + {\overset{ˇ}{b}}_{0}^{(1)} (j) sin (\sqrt{λ^{(0)}} s) + u_{j + \frac{1}{2}, part}^{(1)} (s), s \in [0, 1], j \in N \\ u_{j}^{(1)} (y) = {\overset{ˇ}{c}}_{0}^{(1)} (j) cos (\sqrt{λ^{(0)}} y) + {\overset{ˇ}{d}}_{0}^{(1)} (j) sin (\sqrt{λ^{(0)}} y) + u_{j, part}^{(1)} (y), y \in [0, 1], j \in N^{*} \\ u_{0}^{(1)} (y) = {\overset{ˇ}{c}}_{0, 0}^{(1)} cos (\sqrt{λ^{(0)}} y) + {\overset{ˇ}{d}}_{0, 0}^{(1)} sin (\sqrt{λ^{(0)}} y) + u_{0, part}^{(1)} (y), y \in [0, 1] . \end{matrix} \end{array}$ Let us now determine the above coefficients. First, (74)-(iii) (Neumann b. c.) leads to $\begin{matrix} {\overset{ˇ}{d}}_{0}^{(1)} (j) = 0, \forall j \in Z, {\overset{ˇ}{d}}_{0, 0}^{(1)} = 0 . \end{matrix}$ Next, (74)-(v) (the jump conditions for $j \in N^{*}$ ) give, after some manipulations, $\begin{matrix} (99) & \{\begin{matrix} {\overset{ˇ}{c}}_{0}^{(1)} (j) = \frac{1}{cos (\sqrt{λ^{(0)}} L / 2)} {\overset{ˇ}{a}}_{0}^{(1)} (j) + c_{1} r^{j}, \forall j \in N^{*} & (i) \\ {\overset{ˇ}{b}}_{0}^{(1)} (j) = {\overset{ˇ}{a}}_{0}^{(1)} (j + 1) \frac{1}{sin (\sqrt{λ^{(0)}})} - {\overset{ˇ}{a}}_{0}^{(1)} (j) \frac{1}{tan (\sqrt{λ^{(0)}})} + b_{1} r^{j}, \forall j \in N, & (ii) \end{matrix} \end{matrix}$ where the constants $b_{1}$ and $c_{1}$ are explicit functions of $λ^{(0)}$ (whose expression is omitted on purpose). To be more precise, (99)-(i) directly follows from the first jump condition while (99)-(ii) is deduced from the second jump condition at $j + 1$ taken into account (99)-(i).

Finally, substituting (99) in the fourth equation in (74) (the Kirchhoff condition) yields to $\begin{matrix} (100) & {\overset{ˇ}{a}}_{0}^{(1)} (j + 1) + 2 g (\sqrt{λ^{(0)}}) {\overset{ˇ}{a}}_{0}^{(1)} (j) + {\overset{ˇ}{a}}_{0}^{(1)} (j - 1) = r^{j} a_{1}, \forall j \in N^{*} \end{matrix}$ where, again, $a_{1}$ is an explicit function of $λ^{(0)}$ and g is defined in (13). Using that $r^{2} + 2 g (\sqrt{λ^{(0)}}) r + 1 = 0$ , a direct computation shows that a particular solution of the difference equation (100) is given by $\begin{matrix} (101) & {\overset{ˇ}{a}}_{0, part} (j) = \frac{r a_{1}}{r^{2} - 1} j r^{j}, \forall j \in N . \end{matrix}$ As a result $a (j) : = {\overset{ˇ}{a}}_{0}^{(1)} (j) - {\overset{ˇ}{a}}_{0, part} (j)$ satisfies $\begin{matrix} a (j + 1) + 2 g (\sqrt{λ^{(0)}}) a (j) + a (j - 1) = 0 \forall j \in N^{*} \end{matrix}$ whose general solution in $ℓ_{2} (N)$ has the form $α_{0}^{(1)} r^{j}$ . In other words, we have then proved that $\begin{matrix} (102) & {\overset{ˇ}{a}}_{0}^{(1)} (j) = r^{j} a_{0}^{(1)} (j), with a_{0}^{(1)} (j) : = α_{0}^{(1)} + \frac{r a_{3}}{r^{2} - 1} j, \forall j \in N . \end{matrix}$ Substituting (102) into (99) and (98), we obtain the formulas of the first two lines of (84) where we observe that $\begin{matrix} (103) & a_{ℓ}^{(1)} (j), b_{ℓ}^{(1)} (j), c_{ℓ}^{(1)} (j) and d_{ℓ}^{(1)} (j) are polynomials of degree ⩽ 1 - ℓ with respect to j . \end{matrix}$ Except the constant coefficient of $a_{0}^{(1)} (j)$ (coefficient of the polynomial $a_{0}^{(1)} (j)$ associated with the monomial of degree 0), these polynomials are explicitly determined by formulas (99)–(102). It is consistent with the fact that Problem (74) has a solution up to a multiple of $u^{(0)}$ . At this stage, we can choose $\begin{matrix} (104) & a_{0}^{(1)} (0) = 0, \end{matrix}$ where by definition $a_{0}^{(1)} (0) = u_{1 / 2}^{(1)} (0)$ as our normalisation choice (see Remark 7 with $ℓ : u \mapsto u_{1 / 2} (0)$ ).

It remains to compute ${\overset{ˇ}{c}}_{0, 0}^{(1)}$ for which we use the jump conditions and the Kirchhoff condition for $j = 0$ .

A priori, this gives 3 linear equations in ${\overset{ˇ}{c}}_{0, 0}^{(1)}$ . However, these three equations reduce to a single one allowing us to compute this coefficient explicitely. The verification of this property can be done by hand using the symmetry properties together with the fact that $λ^{(1)}$ is given by (95). However, this property is a consequence of Proposition 5, that ensures that there exists a unique solution to (74) (for $k = 1$ ) satisfying the normalisation condition (104).

4.2.2. Determination of $U^{(1)}$

We already know that $(u^{(0)}, λ^{(0)})$ satisfy (30), (71) and (72). In the same way, as solutions of (74) for $k = 1$ , $(u^{(1)}, λ^{(1)})$ also satisfy (30), (71) and (72).

Since $U^{(0)}$ , obtained by concatenation of the constant functions $U_{j}^{(0)}$ (see (83)) satisfies (35), (53), we have checked that $(H_{1})$ is satisfied. In view of Proposition 4, we can thus assert that, $\begin{matrix} (105) & \forall j \in Z, \exists! U_{j}^{(1)} \in V_{j} solution of (65) for k = 1 satisfying the matching conditions (51–52) . \end{matrix}$ Moreover, $U_{j}^{(1)}$ is shown to be symmetric in the sense of (81) by using the symmetry of $u^{(0)}$ and $u^{(1)}$ . At this stage, we distinguish between the computation of $U_{0}^{(1)}$ and the computation of $U_{j}^{(1)}$ , $j \neq 0$ .

The computation of $U_{0}^{(1)}$ is obtained by solving (65), for $k = 1$ , $j = 0$ , providing the restriction of $U_{0}^{(1)}$ to the junction $K_{0}$ . Then, it is extended to the whole domain $J_{0}$ as in the proof of Proposition 4.

Let us now consider the case $j \neq 0$ and prove (85) by showing that there exists a profile function $U^{(1)} \in V_{j} \equiv V_{1}$ (with $K_{j}$ identified with $K_{1}$ ) and a constant $P^{(1)} (j)$ , that is a polynomial of degree 1 with respect to j, such that the decomposition (85) holds.

We first establish this decomposition inside $K_{j}$ and then extend it to the domain $J_{j}$ as in the proof of Proposition 4, Section 3.2. As for $U_{j}^{(0)}$ , this (straightforward) second step will be omitted.

For $j \in N^{*}$ , according to Section 3.2 and (87), we know that ${\overset{ˇ}{U}}_{j}^{(1)}$ , the restriction of $U_{j}^{(1)}$ in $K_{j}$ satisfies $\begin{matrix} (106) & \{\begin{matrix} Δ {\overset{ˇ}{U}}_{j}^{(1)} = 0 in K_{j}, \\ \partial_{n} {\overset{ˇ}{U}}_{j}^{(1)} + T_{j}^{δ} {\overset{ˇ}{U}}_{j}^{(1)} = r^{j} g_{\pm}^{(δ)} on Σ_{j, δ}, δ = 0, \pm, \\ \partial_{n} U_{j}^{(1)} = 0 on \partial K_{j} ∖ (Σ_{j, \pm} \cup Σ_{j, 0}) \end{matrix} \end{matrix}$ as well as the following condition, which is nothing but the matching conditions (See the electronic version.) $\begin{matrix} \int_{Σ_{j, 0}} U_{j}^{(1)} = u_{j}^{(1)} (- \frac{L}{2}) + \partial_{y} u_{j}^{(0)} (- \frac{L}{2}) . \end{matrix}$ From (82)–(84), one easily finds explicitly a polynomial $P^{(1)}$ of degree 1 such that $\begin{matrix} (107) & \int_{Σ_{j, 0}} U_{j}^{(1)} = r^{j} P^{(1)} (j) . \end{matrix}$ The identification of $K_{j}$ with $K_{1}$ for $j \neq 0$ suggests of course to introduce the profile function ${\overset{ˇ}{U}}^{(1)}$ as the unique solution in $H^{1} (K_{1})$ of (note that the well-posedness of (106) ensures the one of (108))) $\begin{matrix} (108) & \{\begin{matrix} Δ {\overset{ˇ}{U}}^{(1)} = 0 in K_{1}, \\ \partial_{n} {\overset{ˇ}{U}}^{(1)} + T_{j}^{δ} {\overset{ˇ}{U}}^{(1)} = g_{\pm}^{(δ)} on Σ_{j, δ}, δ = 0, +, - (cf. (88)) \\ \partial_{n} {\overset{ˇ}{U}}^{(1)} = 0 on \partial K_{j} ∖ (Σ_{j, \pm} \cup Σ_{j, 0}), \end{matrix} \end{matrix}$ completed with $\int_{Σ_{1, 0}} U^{(1)} = 0$ , So, using (107), by linearity ${\overset{ˇ}{U}}_{j}^{(1)} = r^{j} {\overset{ˇ}{U}}^{(1)} + r^{j} P^{(1)} (j)$ .

4.3. Order k: The general induction step

The previous reasoning can be repeated for any $k ⩾ 2$ . As for $k = 1$ , we shall construct in turn

the coefficient $λ^{(k)}$ and the far field terms $u^{(k)}$ by means of an explicit resolution of the far field problem (74) (see Section 3.3). In particular, we prove that $u^{(k)}$ is symmetric and $\begin{matrix} (109) & \{\begin{matrix} u_{j + \frac{1}{2}}^{(k)} (s) = r^{j} \sum_{ℓ = 0}^{k} s^{ℓ} (a_{ℓ}^{(k)} (j) cos (\sqrt{λ_{0}} s) + b_{ℓ}^{(k)} (j) sin (\sqrt{λ_{0}} s)), s \in [0, 1], j \in N, \\ u_{j}^{(k)} (y) = r^{j} \sum_{ℓ = 0}^{k} y^{ℓ} (c_{ℓ}^{(k)} (j) cos (\sqrt{λ_{0}} y) + d_{ℓ}^{(k)} (j) sin (\sqrt{λ_{0}} y)), y \in [- \frac{L}{2}, 0], j \in N^{*}, \\ u_{0}^{(k)} (y) = \sum_{ℓ = 0}^{k} y^{ℓ} (c_{ℓ, 0}^{(k)} cos (\sqrt{λ_{0}} y) + d_{ℓ, 0}^{(k)} sin (\sqrt{λ_{0}} y)), y \in [- \frac{L}{2}, 0] \end{matrix} \end{matrix}$ the dependence with respect to the parameter j being fully exhibited.

the near field term $U^{(k)}$ . We show that $U^{(k)}$ is symmetric (cf. (81)) and that $\begin{matrix} (110) & U_{j}^{(k)} (\cdot) = r^{j} (\sum_{ℓ = 0}^{k - 1} j^{ℓ} U_{ℓ}^{(k)} (\cdot) + P^{(k)} (j)) j \in N^{*} \end{matrix}$ where $U_{ℓ}^{(k)} \in V_{j} \equiv V_{1}$ (see (34)) are profile functions independent of j and $P^{(k)} (j)$ is a polynomial of degree k with respect to j and constant with respect to $(X, Y) \in J_{j}$ .

We emphasize that the construction, although more technical, is similar to the one for

k = 1

. See the electronic version.

5. Justification of the asymptotic expansion

The existence of the (formal) asymptotic expansion being proved, we now prove Theorem 1 by first constructing (Section 5.1) pseudo-modes as defined in Section 2.1. Then we prove that (19) holds with (18). This is based on error estimates of Sections 5.2 and 5.3. This allows to conclude (Section 5.4).

5.1. Construction of pseudomodes and related properties

Roughly speaking, given $n > 0$ , we construct the approximate far and near fields “of order n” by truncating the expansions (26), (27) and (31). More precisely, for the far field, we define: $\begin{matrix} (111) & \{\begin{matrix} u_{FF}^{ε, n} : Ω_{ε}^{μ} ⟶ R, such that u_{FF}^{ε, n} (x, y) = 0 in K_{j}^{ε, -}, j \in Z and \\ u_{FF}^{ε, n} (x, y) \equiv u_{j + \frac{1}{2}}^{ε, n} (x, y) : = \sum_{k = 0}^{n} ε^{k} u_{j + \frac{1}{2}}^{(k)} (s), s = x - j, (x, y) \in E_{j + \frac{1}{2}}^{-, ε}, j \in Z, \\ u_{FF}^{ε, n} (x, y) \equiv u_{j}^{ε, n} (x, y) : = \sum_{k = 0}^{n} ε^{k} u_{j}^{(k)} (y), (x, y) \in E_{j}^{-, ε}, j \in Z . \end{matrix} \end{matrix}$ In each “thick edge”of $Ω_{ε}^{μ}$ , the couple $(u_{FF}^{ε, n}, λ^{ε, n})$ ( $λ^{ε, n}$ defined in (19), see also below) does not exactly satisfy the desired eigenvalue equation because of the truncation process. More precisely, combining adequately the equations (30) for $0 ⩽ k ⩽ 2 n$ , one finds that (the computations are tedious but straightforward, the details are left to the reader) $\begin{matrix} (112) & Δ u_{FF}^{ε, n} + λ^{ε, n} u_{FF}^{ε, n} = ε^{n + 1} r_{FF}^{ε, n}, in O_{ε}^{μ} : = ⋃_{j \in Z} (E_{j + \frac{1}{2}}^{-, ε} \cup E_{j}^{-, ε}), λ^{ε, n} : = \sum_{k = 0}^{n} ε^{k} λ^{(k)}, \end{matrix}$ where the so-called “far-field remainder” $r_{FF}^{ε, n}$ is such that $r_{FF}^{ε, n} = 0$ in $K_{j}^{ε, -}$ and $\begin{array}{rcl} (113) & \{\begin{matrix} r_{FF}^{ε, n} (x, y) : = \sum_{p = n + 1}^{2 n} ε^{p - n - 1} r_{j + \frac{1}{2}}^{p, n} (x, y), (x, y) \in E_{j + \frac{1}{2}}^{-, ε}, j \in Z, \\ r_{FF}^{ε, n} (x, y) : = \sum_{p = n + 1}^{2 n} ε^{p - n - 1} r_{j}^{p, n} (x, y), (x, y) \in E_{j}^{-, ε}, j \in Z, \\ r_{j + \frac{1}{2}}^{p, n} (x, y) = \sum_{(k, ℓ) \in I^{p, n}} λ^{(k)} u_{j + \frac{1}{2}}^{(ℓ)} (s) s = x - j, r_{j}^{p, n} (x, y) = \sum_{(k, ℓ) \in I^{p, n}} λ^{(k)} u_{j}^{(ℓ)} (y) \end{matrix} \end{array}$ where we have defined, with $I^{p, n} = {(ℓ, k) \in {0, \dots, n}^{2} / ℓ + k = p}$ .

On the other hand, we define for $j \in Z$ the truncated near fields $U_{j}^{ε, n} : J_{j}^{ε} : = (j, 0) + ε J_{j} \to R$ , $\begin{matrix} (114) & U_{j}^{ε, n} (x, y) = \sum_{k = 1}^{n} ε^{k} U_{j}^{(k)} (\frac{x - j}{ε}, \frac{y + L / 2}{ε}) . \end{matrix}$ Again, the couple $(U_{j}^{ε, n}, λ^{ε, n})$ does not exactly satisfy the desired eigenvalue equation in $J_{j}^{ε}$ . Combining adequately the equations (35) for $0 ⩽ k ⩽ 2 n$ , we get (see Remark 10) $\begin{matrix} (115) & Δ U_{j}^{ε, n} + λ^{ε, n} U_{j}^{ε, n} = ε^{n - 1} R_{j}^{ε, n}, in J_{j}^{ε}, j \in Z \end{matrix}$ where, reminding that $I^{p, n} = {(ℓ, k) \in {0, \dots, n}^{2} / ℓ + k = p}$ , the “near-field remainder” $R_{j}^{ε, n}$ is $\begin{matrix} (116) & \{\begin{matrix} R_{j}^{ε, n} (x, y) : = \sum_{p = n - 1}^{2 n} ε^{p - (n - 1)} R_{j}^{p, n} (\frac{x - j}{ε}, \frac{y + L / 2}{ε}), in J_{j}^{ε}, \\ R_{j}^{p, n} : = \sum_{(k, ℓ) \in I^{p, n}} λ^{(k)} U_{j}^{(ℓ)}, in J_{j} . \end{matrix} \end{matrix}$

Remark 10.
The computations that lead to (115) and (116) are of course quite similar to the ones that lead to (112) and (113). The reader will notice that the power of ε that appears as the multiplying factor in the right hand sides of (112) and (115) passes from $n + 1$ in (112) to $n - 1$ (115). Accordingly, there is a difference in the numbers of terms in the sums that define the remainders which passes from $n - 1$ in (113) to $n + 1$ in(116). These changes are due, one on hand to the differences that already occur in equations (30) and (35) respectively, on the other hand to the ε-scaling that appears in the definition (114) of the truncated near fields.

Next, we want to construct a pseudomode that will coincide with $u_{FF}^{ε, n}$ outside some small neighborhood of the junctions $K_{ε, j}^{-}$ and with $U_{j}^{ε, n}$ in a neighborhood of $K_{ε, j}^{-}$ . This will be done in a smooth way with the help of cut-off functions and a partition of unity. Let us first introduce $χ \in C^{\infty} (R)$ such that $\begin{matrix} 0 ⩽ χ (x) ⩽ 1, \forall x \in R, χ (x) = 0, x ⩽ 1, χ (x) = 1, x ⩾ 2, \end{matrix}$ from which we define the 2D cut-off functions (α is a real parameter to be fixed later) $\begin{matrix} χ_{j}^{ε} (x, y) = χ ((x - j) / ε^{α}) χ ((y + L / 2) / ε^{α}), j \in Z, \end{matrix}$ where we impose $0 < α < 1$ : since $α < 1$ , the support of $χ_{j}^{ε}$ is an $ε^{α}$ -neighborhood of $K_{j}^{ε, -}$ and, since $α > 0$ , these supports are disjoint for ε small enough. This allows us to define $\begin{matrix} χ^{ε} = \sum_{j \in Z} χ_{j}^{ε} \end{matrix}$ that coincides which $χ_{j}^{ε}$ on its support. By construction ${(1 - χ^{ε}), {χ_{j}^{ε}, j \in Z}}$ form a partition of unity in $R^{2}$ . The properties of the cut-off functions are illustrated on Figs 7 and 8. Multipying by $(1 - χ^{ε})$ localizes outside the junctions, while multiplying by $χ^{ε}$ (resp. $χ_{j}^{ε}$ ) localizes near the junctions (resp. the jth-junction). In practice, the forthcoming analysis shows that it is advantageous to take α as close as possible to 1 (see estimates (121) and (122)-(ii)), the case $α = 1$ being excluded though (See the electronic version.). This leads us to define the pseudo-mode of order n as follows $\begin{matrix} (117) & u^{ε, n} = (1 - χ^{ε}) u_{FF}^{ε, n} + \sum_{j \in Z} χ_{j}^{ε} U_{j}^{ε, n} . \end{matrix}$ According to Section 2.1, our goal will be to get an estimate of the quantity $\begin{matrix} (118) & I^{ε, n} (v) = \int_{Ω_{ε}^{μ, -}} (\nabla u^{ε, n} \nabla v - λ^{ε, n} u^{ε, n} v) . \end{matrix}$

Fig. 7.
Support of the function $χ^{ε}$ (left) and $1 - χ^{ε}$ (right). White corresponds to 0, black to 1.

Fig. 8.
Support of the function $χ_{j, ε}$ (left) and of its gradient (right).

To obtain a tractable expression $I^{ε, n} (v)$ , we first compute that $\begin{matrix} |\begin{matrix} Δ u^{ε, n} + λ^{ε, n} u^{ε, n} & = (1 - χ^{ε}) (Δ u_{FF}^{ε, n} + λ^{ε, n} u_{FF}^{ε, n}) - 2 \nabla χ^{ε} \cdot \nabla u_{FF}^{ε, n} - Δ χ^{ε} u_{FF}^{ε, n} \\ + \sum_{j \in Z} [χ_{j}^{ε} (Δ U_{j}^{ε, n} + λ^{ε, n} U_{j}^{ε, n}) + 2 \nabla χ_{j}^{ε} \cdot \nabla U_{j}^{ε, n} + Δ χ_{j}^{ε} U_{j}^{ε, n}] . \end{matrix} \end{matrix}$ Using (112), (115) and the fact that, in $supp χ_{j}^{ε}$ , $χ^{ε} \equiv χ_{j}^{ε}$ , this can be rearranged as $\begin{matrix} |\begin{matrix} Δ u^{ε, n} + λ^{ε, n} u^{ε, n} & = ε^{n + 1} (1 - χ^{ε}) r_{FF}^{ε, n} + ε^{n - 1} \sum_{j \in Z} χ_{j}^{ε} R_{j}^{ε, n} \\ - 2 \sum_{j \in Z} \nabla χ_{j}^{ε} \cdot \nabla (u_{FF}^{ε, n} - U_{j}^{ε, n}) - \sum_{j \in Z} Δ χ_{j}^{ε} (u_{FF}^{ε, n} - U_{j}^{ε, n}) . \end{matrix} \end{matrix}$ Multiply the above equality by $- v \in H^{1} (Ω_{ε}^{μ})$ and integrate over $Ω_{ε}^{μ}$ . Using Green’s formula, we get: $\begin{matrix} (119) & I^{ε, n} (v) = - I_{FF}^{ε, n} (v) - I_{NF}^{ε, n} (v) - I_{M}^{ε, n} (v), \end{matrix}$ where by definition $\begin{array}{rcl} (120) & \{\begin{matrix} I_{FF}^{ε, n} (v) = ε^{n + 1} \int_{Ω_{ε}^{μ, -}} (1 - χ^{ε}) r_{FF}^{ε, n} v, I_{NF}^{ε, n} (v) = ε^{n - 1} \sum_{j \in Z} \int_{Ω_{ε}^{μ, -}} χ_{j, ε} R_{j}^{ε, n} v \\ I_{M}^{ε, n} (v) = I_{M, 1}^{ε, n} (v) + I_{M, 2}^{ε, n} (v), \{\begin{matrix} I_{M, 1}^{ε, n} (v) = \sum_{j \in Z} \int_{Ω_{ε}^{μ, -}} \nabla χ_{j}^{ε} \cdot \nabla v (u_{FF}^{ε, n} - U_{j}^{ε, n}), \\ I_{M, 2}^{ε, n} (v) = - \sum_{j \in Z} \int_{Ω_{ε}^{μ, -}} \nabla χ_{j}^{ε} \cdot \nabla (u_{FF}^{ε, n} - U_{j}^{ε, n}) v . \end{matrix} \end{matrix} \end{array}$ In formula (119), we say that:

$I_{FF}^{ε, n} (v)$ is the far field consistency error: it measures how much $(u^{ε, n}, λ^{ε, n})$ fails to satisfy the desired eigenvalue equation inside the support of $1 - χ^{ε}$ ,

$I_{NF}^{ε, n} (v)$ is the near field consistency error: it measures how much $(u^{ε, n}, λ^{ε, n})$ fails to satisfy the desired eigenvalue equation inside the support of $χ^{ε}$ ,

$I_{M}^{ε, n} (v)$ is the matching error: it gathers the mismatch between $u_{FF}^{ε, n}$ and $U_{j}^{ε, n}$ in $supp χ_{j}^{ε}$ for all j.
5.2. Estimation of the matching error

Lemma 5.
The matching error $I_{M}^{ε, n} (v)$ defined by ( 120 ) satisfies the estimate: $\begin{matrix} (121) & | I_{M}^{ε, n} (v) | ⩽ C_{α, n} ε^{\frac{1}{2}} ε^{α n} ‖ v ‖_{H^{1} (Ω_{ε}^{μ})} . \end{matrix}$
Proof.
See the electronic version. □
5.3. Estimation of the consistency errors

Lemma 6.
The consistency errors defined in ( 120 ) satisfy the estimates $\begin{matrix} (122) & (i) | I_{FF}^{ε, n} (v) | ⩽ C_{n} ε^{n + \frac{3}{2}} ‖ v ‖_{L^{2} (Ω_{ε}^{μ})} (ii) | I_{NF}^{ε, n} (v) | ⩽ C_{n} ε^{(n + 1) α - \frac{1}{2}} ‖ v ‖_{H^{1} (Ω_{ε}^{μ, -})} . \end{matrix}$
Proof.
See the electronic version. □
5.4. Completion of the proof of Theorem 1

Using (with m instead of n) the estimates (121) (Lemma 5) and (122) (Lemma 6), we deduce from (119) (with m instead of n) that the estimate (19) holds with $α_{m} = α m + α - \frac{1}{2}$ . According to Lemma 1, the proof of Theorem 1 is thus complete.

6. A numerical approach based on asymptotics expansions

6.1. Description of the method

See the electronic version.

6.2. Numerical results

In the following section, we choose $L = 2$ . In that case, the essential spectrum of the limit operator $A_{s}^{μ}$ is given by (see Proposition 4 and Fig. 9 in [5]) $\begin{array}{l} σ_{ess} (A_{s}^{μ}) = {λ = ω^{2} \in R^{+}, ω \in ⋃_{k \in N} I_{k}} where \\ I_{k} = [- arccos (\frac{1}{3}) + k π, arccos (\frac{1}{3}) + k π] \forall k \in N, \end{array}$ while, for any $μ < 1$ , the discrete spectrum of $A_{s}^{μ}$ is $\begin{matrix} σ_{d} (A_{s}^{μ}) = {λ = ω^{2}, ω \in ⋃_{k \in N} {λ_{*} (μ) + k π, - λ_{*} (μ) + k π}}, \end{matrix}$ where $λ_{*} (μ)$ denotes the unique root of the equation (9-right) in $[0, \frac{π}{2}]$ . To obtain $σ_{d} (A_{s}^{μ})$ , we used Theorem 1 in [5] ensuring that $A_{s}^{μ}$ has exactly two eigenvalues in each of its gaps together with the fact that if $λ = ω^{2}$ is a solution of (9), both $λ^{'} = {(ω + k π)}^{2}$ and $λ^{″} = {(k π - ω)}^{2}$ , for any $k \in N$ are also solutions of (9). We notice that the spectrum of $A_{s}^{μ}$ is the image by the function $x \mapsto x^{2}$ of a π-periodic subspace of $R$ . A graphic illustration of $σ (A_{s}^{μ})$ is presented in Fig. 9.

Fig. 9.

The spectrum of $σ (A_{s}^{μ})$ in the case $L = 2$ .

Computation of eigenvalues and numerical validation of the method. In the following experiments, we take $μ = 1 / 4$ and we focus on the first and fourth eigenvalues $λ_{1} = λ_{*} (\frac{1}{4}) \approx 1.80$ and $λ_{4} \approx 24.43$ , located respectively in the first and second gaps of the operator $A_{s}^{1 / 4}$ (see Fig. 9). The associated limit eigenvectors (defined on the graph by (23–24)) are represented on Fig. 10.

Fig. 10.

Representation of two limit eigenvectors associated with $λ_{1}$ and $λ_{2}$ .

The numerical results associated with the first eigenvalue $λ_{1}$ are represented on Fig. 11. In the left part, we compute the evolution of $\begin{matrix} (123) & λ^{ε, n} = \sum_{k = 0}^{n} ε^{k} λ^{(k)} \end{matrix}$ with respect to ε for n varying between 1 and 5 and ε between 0.02 and 0.6. To compute the near field terms (part 3-(c) of the algorithm described in Section 6.1), we first truncate the junctions $J_{0}$ and $J_{1}$ at a distance $T = 5$ and we use a first order approximation of the Dirichlet-to-Neumann operator. The problem is then numerically solved by a $P_{1}$ -finite element method using a uniform mesh of mesh-size $h = 0.002$ . We compare $λ^{ε, n}$ with a reference value of $λ^{ε}$ obtained by computing numerically the first eigenvalue of the full two dimensional operator $A_{ε, s}^{μ}$ using the method developed in [9]. In a nutshell, this method permits us to rewrite the initial eigenvalue problem, posed in the unbounded domain $Ω_{ε}^{μ}$ , as a non linear eigenvalue problem posed in a bounded domain. This is done by computing the (approximate) Dirichlet-to-Neumann operators for periodic domains (see [8,12]), which requires to solve periodic cell problems (discretized here again using the standard $P_{1}$ finite element methods) and a stationary Ricatti equation. By contrast to the initial problem, the reduced problem (posed in a bounded domain) is a non linear eigenvalue problem (since the DtN operators depend on the eigenvalue) of a fixed point nature. It is solved using a Newton-type procedure, each iteration needing a finite element computation, see [9] for more details.

We notice that the approximation of $λ^{ε}$ by $λ^{ε, n}$ is qualitatively good, especially when adding high order terms in the truncated series (123). Surprisingly, the approximation remains accurate even for a rather large ε (the geometry of the domain $Ω_{ε}^{μ}$ for $ε = 0.6$ does not really looks like a graph-like structure).

Fig. 11.

Results for the first eigenvalue: $λ^{ε, 1}$ w.r.t ε (left), error w.r.t ε (right).

Fig. 12.

Results for the fourth eigenvalue:: $λ^{ε, 4}$ w.r.t ε (left), error w.r.t ε (right).

To verify the accuracy of our asymptotic expansion, we represent on Fig. 11 the evolution of the errors $e_{n} = | λ^{ε, n} - λ^{ε} |$ with respect to ε, ε varying between 0.02 and 0.6. For the first two orders, the experimental convergence rates (2.1 for $e_{1}$ and 2.9 for $e_{2}$ ) coincide with the theoretical ones. Unfortunately, this is not the case for the higher order ones. It might be due to the fact that the ‘exact’ solution $λ^{ε}$ is computed with a limited precision of $10^{- 3}$ . The use of a second order finite element method for the different numerical computations may confirm this point but is beyond the scope of this paper.

The same experiment is reproduced for the fourth eigenvalue ( $λ_{4} \approx 24.43$ ) on Fig. 12. Here again, the approximation of $λ^{ε}$ by $λ^{ε, n}$ is qualitatively good, especially for high orders. However, we observe that for a given ε the error is bigger for $λ_{4}$ than for $λ_{1}$ . We point out that we are not able to compute $λ^{ε}$ for $ε > 0.25$ . In that case, we do not know if the eigenvalue is close to the essential spectrum or does not exist anymore.

To summarize, from a computational point of view, the main advantage of the asymptotic method is that it suffices to make one computation in order to obtain an approximation of $λ^{ε}$ for an arbitrary value ε. Moreover, the approximation is highly-accurate when ε is small (the accuracy depending of the numerical error made in the computation of the near field terms). Nevertheless, by nature, the asymptotic method fails to predict the possible disparition of the eigenvalue into the essential spectrum as ε becomes large. In that case, a high order direct method would be preferable (see e.g [15]).

Fig. 13.

Schematic representation of the domain under consideration.

Fig. 14.

Near fields $U_{0}^{(1)}$ (left) and $U_{0}^{(2)}$ (right) in $J_{0}$ . Bottom picture: zoom inside the white rectangle.

Example of near field terms. To end this part, let us give a few examples of near field terms obtained when computing the first eigenvalue $λ_{1}$ (see Fig. 9). We shall focus on the junctions $J_{0}$ ( $j = 0$ , perturbed junction) and $J_{1}$ ( $j = 1$ , first unmodified junction) as represented on Fig. 13.

Fig. 15.

Cuts of the near fields $U_{0}^{(1)}$ and $U_{0}^{(2)}$ along $y = \frac{1}{2}$ (left) and $x = 0$ in $J_{0}$ (right).

Fig. 16.

Near fields $U_{1}^{(1)}$ (left) and $U_{1}^{(2)}$ (right) in $J_{1}$ . Bottom picture: zoom inside the white rectangle.

Fig. 17.

Cuts of the near fields $U_{0}^{(1)}$ and $U_{0}^{(2)}$ along $y = \frac{1}{2}$ (left) and $x = 0$ in $J_{1}$ (right).

In Fig. 14, we display the near fields terms $U_{0}^{(1)}$ and $U_{0}^{(2)}$ , that is to say the near field terms of order 1 and 2 in the junction $J_{0}$ . A zoom on the central part of the junction is represented on the right part of the two pictures. We remark that these two fields are symmetric with respect to $x = 0$ . Moreover, they tend to grow inside the branches of the junction. To quantify this growth, we plot on Fig. 15 the representative curves of the two fields along the horizontal cut $y = 1 / 2$ and the vertical cut $x = 0$ . As expected, the first near field is linearly increasing while the second one has a quadratic growth.

Then, the same near fields are represented in the junction $J_{1}$ are displayed on Figs 16–17. As expected, the fields are not symmetric anymore. However, they are still polynomial growing, but the polynomials are different in the left ( $x < 0$ ) and right part ( $x > 0$ ) of the junction.

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Trapped modes in thin and infinite ladder like domains. Part 2: Asymptotic analysis and numerical application

Abstract

Keywords

1. Summary of Part 1 and main results of Part 2

1.1.1. The operator A ε μ

1.1.2. The decomposition of the operator A ε μ into its symmetric and antisymmetric components

1.2. The limit problem: Spectral problem on the graph

1.2.1. The limit operators A μ , A s μ and A a μ

2.1. Methodology of the proof

2.3.1. Derivation of the matching conditions

3.1. Necessary conditions for the existence of the near field terms

3.1.1. Dirichlet to Neumann operators

3.1.2. Reduction to a bounded domain

Proposition 4. Assume that ( H k ) holds. Then, for each j, there exists a unique near field term U j ( k ) ∈ V j satisfying ( 35 ) and ( 55 ). Proof. See the electronic version. □ 3.3. The problem in ( u ( k ) , λ ( k ) )

4.1. Order 0: Initialization of the algorithm

4.2. Order 1: First induction step

4.2.2. Determination of U ( 1 )

4.3. Order k: The general induction step

5. Justification of the asymptotic expansion

5.1. Construction of pseudomodes and related properties

Lemma 5. The matching error I M ε , n ( v ) defined by ( 120 ) satisfies the estimate: (121) | I M ε , n ( v ) | ⩽ C α , n ε 1 2 ε α n ‖ v ‖ H 1 ( Ω ε μ ) . Proof. See the electronic version. □ 5.3. Estimation of the consistency errors

6. A numerical approach based on asymptotics expansions

6.1. Description of the method

6.2. Numerical results

References

1.1.1. The operator $A_{ε}^{μ}$

1.1.2. The decomposition of the operator $A_{ε}^{μ}$ into its symmetric and antisymmetric components

1.2.1. The limit operators $A^{μ}$ , $A_{s}^{μ}$ and $A_{a}^{μ}$

Proposition 4.
Assume that $(H_{k})$ holds. Then, for each j, there exists a unique near field term $U_{j}^{(k)} \in V_{j}$ satisfying ( 35 ) and ( 55 ).
Proof.
See the electronic version. □

3.3. The problem in $(u^{(k)}, λ^{(k)})$

4.2.2. Determination of $U^{(1)}$