Hélène Barucq

Aralar Erdozain

Victor Péron

Impedance Transmission Conditions for the Electric Potential across a Highly Conductive Casing

Keywords: Asymptotic Models, Impedance Conditions, Electric Potential, Borehole, Casing, Resistivity, Conductivity Modèles Asymptotiques, Conditions d'Impedance, Potentiel Électrique, Forage, tube, Résistivité, Conductivité

Borehole resistivity measurements are a common procedure when trying to obtain a better characterization of the Earth's subsurface. The use of a casing surrounding the borehole highly complicates the numerical simulations due to a large contrast between the conductivities of the casing and the rock formations. In this work, we consider the casing to be a thin layer of uniform thickness and motivated by realistic scenarios, we assume that the conductivity of such casing is proportional to the thickness of the casing to the power of -3. We derive Impedance Transmission Conditions (ITCs) for the static (zero frequency) electric potential for a 2D conguration. Then, we analyse these models by proving stability and convergence results. Next, we asses the numerical performance of these models by employing a Finite Element Method. Finally we present present asymptotic models for similar congurations including the time-harmonic conguration and a 3D axisymmetric scenario.

Introduction

Resistivity measurements are commonly used when trying to obtain a better characterization of the earth's subsurface. The standard procedure for acquiring these resistivity measurements consists in employing one or several transmitters and receiver antennas. These transmitters and receivers, which we will refer to as the instrument, are located inside a well, then they are employed both to transmit electromagnetic waves into the layer formations and record the received waves. According to the results shown in [START_REF] Kaufman | The electrical eld in a borehole with a casing[END_REF][START_REF] Kaufman | A transmission-line model for electrical logging through casing[END_REF], the second derivative of electric potential in the vertical direction, measured at the receiving antennas, can be employed to determine the conductivity of the dierent layer formations composing the Earth's subsurface. This technique has been widely employed in the literature for acquiring borehole resistivity measurements, we refer the reader to [START_REF] Pardo | A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: electrostatics[END_REF][START_REF] Pardo | Simulation of multifrequency borehole resistivity measurements through metal casing using a goal-oriented hp nite-element method[END_REF][START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF][START_REF] Pardo | PML enhanced with a self-adaptive goal-oriented hp-nite element method: Simulation of through-casing borehole resistivity measurements[END_REF][START_REF] Pardo | Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp nite-element method. Part II: through-casing resistivity instruments[END_REF][START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF][START_REF] Chen | New post-processing method for interpretation of through casing resistivity (TCR) measurements[END_REF][START_REF] Qing | Compensation eect analysis in DIE method for through-casing measuring formation resistivity[END_REF] for more information regarding this matter.

Electrical logging through casing is of special interest because the well is often surrounded by a steel-made casing. The use of such casings allows to protect the well and avoid possible collapses on one hand, but it also highly complicates the numerical simulations for the electric potential on the other hand due to the high conductivity and thinness of the casing compared to that of the layer formations. Thus, when performing these kind of studies, the results are often inaccurate or simply too costly to be performed in real time.

These kind of problems have been faced by two dierent approaches, the use of analytical methods and the use of numerical methods. The use of analytical methods [START_REF] Muga | Semi-analytical response of acoustic logging measurements in frequency domain[END_REF][START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF][START_REF] Pérez-Arancibia | On the Green's function for the Helmholtz operator in an impedance circular cylindrical waveguide[END_REF] limits the types of geometries that one can consider, so it is not very suitable for modelling realistic physical congurations. The use of numerical methods seems the best answer for dealing with complex geometries. A wide range of techniques can be found regarding the numerical methods. The Discontinuous Petrov-Galerking method [START_REF] Demkowicz | A class of discontinuous PetrovGalerkin methods. Part I: The transport equation[END_REF][START_REF] Zitelli | A class of discontinuous PetrovGalerkin methods. Part IV: The optimal test norm and timeharmonic wave propagation in 1D[END_REF], the Isogeometric analysis [START_REF] Hughes | Isogeometric analysis: CAD, nite elements, NURBS, exact geometry and mesh renement[END_REF][START_REF] Schillinger | Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis[END_REF] and the hp-adaptive Finite Element Method [START_REF] Pardo | A self-adaptive goaloriented hp-nite element method with electromagnetic applications[END_REF][START_REF] Pardo | Two-Dimensional High-Accuracy Simulation of Resistivity Log ging-While-Drilling (LWD) Measurements Using a Self-Adaptive Goal-Oriented hp Finite Element Method[END_REF][START_REF] Pardo | A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: electrostatics[END_REF][START_REF] Solin | Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell's equations[END_REF] are examples of techniques which are worth mentioning. However, this option becomes challenging too due to the high electrical conductivity contrast between the metallic casing and the layer formations, as well as the small thickness of the casing. In particular, when dealing with this kind of thin layers, the computational cost greatly increases when trying to mesh it. Moreover, the numerical methods employed to solve these problems do not perform well when high contrasted media is considered. These facts lead to an unavoidable increase of the computational cost, so it is relevant to avoid the thin layer by employing mathematical techniques which allow to construct reduced problems involving appropriate boundary or transmission conditions

To overcome these diculties we adopt an asymptotic method which is motivated by realistic congurations [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF], where the conductivity in the casing takes much higher values than that in the layer formations. We intend to work in the context of this application for which we assume that the conductivity in the casing has the following form

σ lay ≈ ε -3 ,
where ε denotes the thickness of the casing, which is presented as a thin layer of uniform thickness. We can motivate this choice according to the paper [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF], where we can observe the following values for the conductivity and the thickness of the casing

   ε = 1.27 • 10 -2 m, σ lay = 4.34 • 10 6 Ω -1 m -1 .
From these values we infer the following relation between these physical parameters σ lay = 8.89 • ε -3 .

In this framework our aim is to derive Impedance Transmission Conditions (ITCs) for the electric potential across such a casing. The naturally small thickness of the casing compared to the rest of the domain makes it ideal for applying this kind of method. The concept of Impedance Conditions (ICs) and ITCs is classical in the modelling of wave propagation phenomena, such a condition is derived by performing an asymptotic expansion and is designed to replace one part of the computational domain. Asymptotic techniques are widely employed in the eld of wave propagation problems, for instance we cite the works [START_REF] Caloz | Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism[END_REF][START_REF] Caloz | On the inuence of the geometry on skin eect in electromagnetism[END_REF][START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF][START_REF] Haddar | Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers[END_REF][START_REF] Buret | Eddy currents and corner singularities[END_REF][START_REF] Ida | Impedance boundary conditions for transient scattering problems[END_REF] related to boundary layer phenomena in Electromagnetism (skin eect and eddy current problem).

Similar studies regarding the derivation ICs for electromagnetism include [START_REF] Chun | High-order accurate thin layer approximations for timedomain electromagnetics. Part I: General metal backed coatings[END_REF][START_REF] Poignard | Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer[END_REF][START_REF] Poignard | Approximate transmission conditions through a weakly oscillating thin layer[END_REF][START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF][START_REF] Bendali | The eect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation[END_REF][START_REF] Senior | Generalized impedance boundary conditions in scattering[END_REF] where ICs are derived to substitute a thin layer present in one border of the domain. The question of ITCs is more related to the present work, but it is also more delicate than that of ICs. Even so, we can also nd a wide variety of works related to this topic, [START_REF] Durué | Thin layer models for electromagnetism[END_REF][START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF][START_REF] Delourme | Modèles et asymptotiques des interfaces nes et périodiques en électromagnétisme[END_REF][START_REF] Chun | High-order accurate thin layer approximations for time-domain electromagnetics. Part II: transmission layers[END_REF][START_REF] Poignard | Approximate transmission conditions through a weakly oscillating thin layer[END_REF][START_REF] Péron | Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer[END_REF][START_REF] Engquist | Eective boundary conditions for acoustic and electromagnetic scattering in thin layers[END_REF][START_REF] Schmidt | A unied analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model[END_REF][START_REF] Schmidt | High order transmission conditions for thin conductive sheets in magneto-quasistatics[END_REF][START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF][START_REF] Poignard | Rigorous asymptotics for the elecric eld in TM mode at mid-frequency in a bidimensional medium with thin layer[END_REF][START_REF] Péron | Mathematical modeling of electromagnetic phenomena in high contrast media[END_REF][START_REF] Schmidt | High-order numerical modelling of highly conductive thin sheets[END_REF].

This study is performed in the framework of high contrasted media where the physical parameters have a dependence on the thickness of the thin layer. Several works can be found with similarities in this matter, for instance, in [START_REF] Schmidt | Robust transmission conditions of high order for thin conducting sheets in two dimensions[END_REF], the authors derive ITCs for eddy current models with a dependence on the conductivity parameter of the thin layer of the form ε -1 and ε -2 . In the same way, in [START_REF] Péron | Equivalent transmission conditions for the timeharmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet[END_REF], we nd a thin layer problem for the time-harmonic Maxwell equations, whose conductivity depends on the thickness of the thin layer in the form of ε -2 . In [START_REF] Perrussel | Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer[END_REF] and [START_REF] Durué | Thin layer models for electromagnetism[END_REF], where a problem for the static potential and an electromagnetic problem is considered respectively, and in both works a resistive thin layer is present.

There exist also similar studies regarding the derivation of ITCs for other physical models, for example we can mention [START_REF] Burel | Contributions à la simulation numérique en élastodynamique: découplage des ondes P et S, modèles asymptotiques pour la traversée de couches minces[END_REF] regarding the study of Elastodynamics, [START_REF] Péron | Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer[END_REF][START_REF] Péron | Equivalent boundary conditions for an elasto-acoustic problem set in a domain with a thin layer[END_REF] regarding the study of a problem with elastic and acoustic media and [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] in the eld of Acoustics. There exist also models where the physical parameters depend on the thickness of the thin layer, [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF] perfoms a study about the problem of an elastic shell-like inclusion with a rigidity of the form ε -1 and ε - 3 .

In this work, we consider non-smooth computational domains, which include vertices and edges. In general, this framework greatly complicates the analysis compared to the smooth case (see for example [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF]) and the presence of geometrical singularities (such as corners) reduces the performance of standard impedance conditions, see for example [START_REF] Buret | Eddy currents and corner singularities[END_REF][START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF][START_REF] Vial | Analyse multi-échelle et conditions aux limites approchées pour un problème avec couche mince dans un domaine à coin[END_REF][START_REF] Vial | Eciency of approximate boundary conditions for corner domains coated with thin layers[END_REF]. In this work, we consider mainly a transmission problem for the electric potential div [(σ -i 0 ω) ∇u] = f in Ω, with an homogeneous Dirichlet boundary condition. We rst consider the domain Ω to be a rectangular shaped domain in R 2 , and then we consider the domain Ω to be an axisymmetric borehole shaped domain in R 3 . This domain is composed of three subdomains Ω ε int , Ω ε ext and Ω ε lay , where the last one corresponds to the casing and is a thin layer of uniform thickness. Here ω represents the frequency and 0 represents the permittivity. The parameter σ corresponds to the conductivity and it is a piecewise constant function that takes dierent values in each subdomain. The function f corresponds to the right-hand side and it is a function that vanishes in the casing.

In this framework, we address the issue of ITCs for u as ε tends to zero. We derive two dierent classes of ITCs employing dierent approaches. The rst one consists in deriving ITCs across the casing itself, whereas the second approach tackles the problem by deriving ITCs on an articial interface located in the middle of the casing. Both classes have their advantages and drawbacks, matter which we shall discuss in this work. We shall present the mathematical justication for these ITCs and we shall also concentrate on studying the numerical performance of the models derived. First class of ITCs appear as second and fourth order approximations whereas second class of ITCs are derived from order 1, up to fourth order 4.

The asymptotic method we follow can be summarized in the following steps. First of all we perform a scaling in the subdomain corresponding to the casing, Ω ε lay , along the direction normal to the thin layer. Then we perform an Ansatz in form of power expansion of ε and we obtain a collection of problems. They can be alternately solved to determine the elementary problems satised by each term of the asymptotic expansion. Then we truncate the series and collect the rst terms of the expansion to infer equivalent conditions by neglecting residual terms depending on ε. Finally we prove convergence results for the derived asymptotic models. We follow this methodology for both the 2D and the 3D congurations. This document is divided in several sections. We begin by presenting the model problem we are interested in, we study the well posedness of this problem and we present the rst step towards the derivation of asymptotic models. Sections 2 and 3 are devoted to the formal calculi of the derivation of the asymptotic models for the two dierent classes. Section 4 corresponds to the validation of the asymptotic models derived for both the rst and second classes. Then, Section 5 presents some numerical results to illustrate the theoretical results of the previous sections, and nally, the last section corresponds to an appendix summarizing similar results for other congurations of interest, including a time-harmonic problem and a 3D axisymmetric scenario.

1

Model problem and scaling

This section acts as an introduction to the following two sections. Here, we present a transmission problem for the electric potential in a conguration where a highly conductive thin layer of uniform thickness ε is present. Later on, in the following two sections, we will employ this model problem for deriving two dierent classes of ITCs employing two dierent approaches which correspond to these two sections.

In this section we rst present the model problem we are interested in studying. Then, we begin the process of deriving asymptotic models by performing the rst step of the procedure, which corresponds to a scaling in the part of the domain corresponding to the casing. This step is common to both approaches we have considered. Let Ω ⊂ R 2 be the domain of interest described in Figure 1. The domain Ω is rectangular shaped and is composed of three rectangular shaped subdomains Ω ε int , Ω ε ext and Ω ε lay . The subdomain Ω ε lay is a thin layer of uniform thickness ε > 0. We denote the interfaces the interface between Ω ε int and Ω ε lay by Γ ε int , and the interface between Ω ε lay and Ω ε ext by Γ ε ext . In this domain, we study the equations of the static electric potential, which read as follows

Ω ε lay Ω ε int Ω ε ext Γ Γ ε int Γ ε ext ε x 0 L y 0 x y n n
div(σ∇u) = f, (1)
along with an homogeneous Dirichlet boundary condition. Here, u represents the electric potential, σ is the conductivity and f stands for a current source. The conductivity is a piecewise constant function, with a dierent value in each subdomain. Specically, the value of the conductivity inside the thin layer Ω ε lay is much larger than the one in the other subdomains and Inria we assume that it depends on the parameter ε. We consider a conductivity of the following form

σ =          σ int in Ω ε int , σ lay = σ 0 ε -3 in Ω ε lay , σ ext in Ω ε ext ,
where σ 0 > 0 is a given constant. We assume that the right-hand side f , is a piecewise smooth function independent of ε, and it vanishes inside the layer.

f =          f int in Ω ε int , f lay = 0 in Ω ε lay , f ext in Ω ε ext .
It is possible to prove that Problem (1) has a unique solution u ∈ H 1 0 (Ω). Then, representing the solution u in each subdomain as follows

u =          u int in Ω ε int , u lay in Ω ε lay , u ext in Ω ε ext , the Problem (1) reads as                                            σ int ∆u int = f int in Ω ε int , σ ext ∆u ext = f ext in Ω ε ext , ∆u lay = 0 in Ω ε lay , u int = u lay on Γ ε int , u lay = u ext on Γ ε ext , σ int ∂ n u int = σ 0 ε -3 ∂ n u lay on Γ ε int , σ 0 ε -3 ∂ n u lay = σ ext ∂ n u ext on Γ ε ext , u = 0 on ∂Ω, (2)
where ∂ n represents the normal derivative in the direction of the normal vector, inwardly directed to Ω ε ext on Γ ε ext , and outwardly directed to Ω ε int on Γ ε int , see Figure 1.

Remark 1. In this document we present the results for a problem where we consider homogeneous Dirichlet boundary conditions. Similar results can be found in [START_REF] Erdozain | Fast inversion of 3D Borehole Resistivity Measurements using Model Reduction Techniques based on 1D Semi-Analytical Solutions[END_REF] for a problem where mixed (Dirichlet and Neumann) conditions are considered, conguration which is more relevant towards the applications.

First of all, before starting with the derivation of a multiscale expansion, we will prove that there exists a solution to Problem [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF] and that this solution is unique in H 1 0 (Ω). Instead of considering directly Problem (2), we will consider a similar one. This problem is dened employing the same conguration we have dened in this section. We remember that the constants σ int , σ ext and σ 0 are strictly positive as this fact will play an important role in the following proofs. We recall Figure 1 shows the conguration of the domain we are working with. In this framework we consider the following problem

                                           σ int ∆u int = f int in Ω ε int , σ ext ∆u ext = f ext in Ω ε ext , σ 0 ε -3 ∆u lay = f lay in Ω ε lay , u int = u lay on Γ ε int , u lay = u ext on Γ ε ext , σ 0 ε -3 ∂ n u lay -σ int ∂ n u int = g int on Γ ε int , σ 0 ε -3 ∂ n u lay -σ ext ∂ n u ext = g ext on Γ ε ext , u = 0 on ∂Ω. ( 3
)
This problem is similar to Problem (2) and it generalises it as the right-hand side f lay does not vanish inside the layer any more and as it includes the right-hand side functions g int and g ext . The results obtained for this problem will be useful in later sections, where we prove the convergence of the asymptotic models.

First of all we can write the variational formulation of Problem [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF].

Assuming f ∈ L 2 (Ω), g int ∈ L 2 (Γ ε int ) and g ext ∈ L 2 (Γ ε ext ), we look for u ∈ H 1 0 (Ω), such that for all w ∈ H 1 0 (Ω) a(u, w) = l(w) (4)
where

a(u, w) = σ int Ω ε int ∇u • ∇w dx + σ ext Ω ε ext ∇u • ∇w dx + σ 0 ε -3 Ω ε lay ∇u • ∇w dx, l(w) = - Ω ε int f int w dx - Ω ε ext f ext w dx - Ω ε lay f lay w dx + Γ ε int g int w ds + Γ ε ext g ext w ds.
Now we give the theorem and proof that guarantees the existence and uniqueness of a solution to this problem and which presents some uniform estimates for the solution, but before we will present the following notation for the dierent norms we employ in this document. Notation 1. For any function u ∈ L 2 (Ω), we denote the norm in L 2 by

u 0,Ω = u L 2 (Ω) .
In the same way, for any function u ∈ H 1 (Ω), we denote the norm in H 1 by

u 1,Ω = u 2 0,Ω + ∇u 2 0,Ω 1 2 .
Inria Theorem 1. For all ε > 0 there exists a unique u ∈ H 1 0 (Ω), solution to Problem [START_REF] Burel | Contributions à la simulation numérique en élastodynamique: découplage des ondes P et S, modèles asymptotiques pour la traversée de couches minces[END_REF] with data

f ∈ L 2 (Ω), g int ∈ L 2 (Γ ε int ), g ext ∈ L 2 (Γ ε ext ).
Moreover, there exists ε 0 > 0 and a constant C > 0, such that for all ε ∈ (0, ε 0 ),

u 1,Ω ≤ C f 0,Ω + g int 0,Γ ε int + g ext 0,Γ ε ext .
Proof. The proof can be summarized in the following steps. As the bilinear form a is coercive and continuous in H 1 0 (Ω), and the linear form l is continuous in H 1 0 (Ω), the existence and uniqueness of a weak solution follows from the Lax-Milgram Lemma. Then, for proving the uniform estimates, we select ε 0 as

ε 0 = 3 σ 0 min(σ int , σ ext )
.

Then for ε < ε 0 and employing the Poincaré inequality, there exists a constant

k 1 > 0 such that a(w, w) ≥ 1 k 1 min (σ int , σ ext ) w 2 1,Ω . (5)
Applying a trace theorem and Cauchy-Schwarz inequality to the denition of l we obtain that for a constant k 2 > 0, we have

|l(w)| ≤ k 2 w 1,Ω f int 0,Ω ε int + f ext 0,Ω ε ext + f lay 0,Ω ε lay + g int 0,Γ ε int + g ext 0,Γ ε ext . (6)
Finally, employing Equations ( 5) and ( 6) we obtain

u 1,Ω ≤ C f int 0,Ω ε int + f ext 0,Ω ε ext + f lay 0,Ω ε lay + g int 0,Γ ε int + g ext 0,Γ ε ext , where C = k 2 k 1 min (σ int , σ ext )
.

Introduction of a scaling

A key point for the derivation of a multiscale expansion for the solution to Problem (2) consists in performing a scaling along the normal direction to the thin layer. We begin by describing the domain Ω ε lay in the following way

Ω ε lay = γ(y) + εXn : γ(y) ∈ Γ, X ∈ - 1 2 , 1 2
,
where γ is a parametrization of the curve Γ (see Figure 1), which is dened as γ(y) = (x 0 , y), for all y ∈ (0, y 0 )

and n = (1, 0) is the normal vector to the curve Γ. This geometry of the domain induces the following scaling

x = x 0 + εX ⇔ X = ε -1 (x -x 0 ) .
As a consequence, we have

RR n°8998 ∂ k X = ε k ∂ k x , k ∈ N.
This scaling allows us to write the Laplace operator in the following way

∆ = ∂ 2 x + ∂ 2 y = ε -2 ∂ 2 X + ∂ 2 y .
Besides, we notice that on the interfaces Γ ε int and Γ ε ext we can rewrite the normal derivative in the following form

∂ n = ∂ x = ε -1 ∂ X .
Finally we denote by U the function that meets

u lay (x, y) = u lay (x 0 + εX, y) = U (X, y), (X, y) ∈ - 1 2 , 1 2 × (0, y 0 ) .
We rewrite Equations ( 2) with the newly dened variables and functions and they take the following form

                                                                 σ int ∆u int = f int in Ω ε int , σ ext ∆u ext = f ext in Ω ε ext , ε -2 ∂ 2 X U + ∂ 2 y U = 0 in -1 2 , 1 2 × (0, y 0 ) , u int x 0 - ε 2 , y = U - 1 2
, y y ∈ (0, y 0 ),

u ext x 0 + ε 2 , y = U 1 2 , y y ∈ (0, y 0 ), σ int ∂ n u int x 0 - ε 2 , y = σ 0 ε -4 ∂ X U - 1 2
, y y ∈ (0, y 0 ),

σ ext ∂ n u ext x 0 + ε 2 , y = σ 0 ε -4 ∂ X U 1 2 , y y ∈ (0, y 0 ), u = 0 on ∂Ω. (7)
Inria 2

First class of ITCs

The objective of this section is the derivation of asymptotic models for the reference model [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF]. In Section 2.1 we make an asymptotic expansion of the solution in power series of ε. Then, by truncating this series and neglecting higher order terms in ε, we derive approximate models composed of equivalent transmission conditions across the thin layer in Section 2.2. The last section, Section 2.3, shows a comparison of the models derived here with more standard models.

Construction of a multiscale expansion

First of all, we dene the jump and mean value of a function across the thin layer.

Denition 1. Let u be a smooth function dened over Ω. We dene its jump and mean value across the thin layer as

     [u] Γ ε = u ext | Γ ε ext -u int | Γ ε int , {u} Γ ε = 1 2 u ext | Γ ε ext + u int | Γ ε int .
Now we proceed to derive the asymptotic expansion. To begin with, we perform an Ansatz in the form of power series of ε for the solution to Problem [START_REF] Caloz | On the inuence of the geometry on skin eect in electromagnetism[END_REF]. We look for solutions

                       u int (x, y) ≈ k≥0 ε k u k int (x, y) in Ω ε int , u ext (x, y) ≈ k≥0 ε k u k ext (x, y) in Ω ε ext , U (X, y) ≈ k≥0 ε k U k (X, y) in - 1 2 , 1 2 × (0, y 0 ) . (8)

Equations for the coecients of the electric potential

Substituting the previous expressions (8) into the Equations ( 7) and collecting the terms with the same powers in ε, for every k ∈ N we obtain the following set of equations

             σ int ∆u k int (x, y) = f int (x, y)δ k 0 in Ω ε ext , σ ext ∆u k ext (x, y) = f ext (x, y)δ k 0 in Ω ε ext , ∂ 2 X U k (X, y) = -∂ 2 y U k-2 (X, y) in - 1 2 , 1 2 × (0, y 0 ) , (9)
along with the following transmission conditions

RR n°8998                                U k - 1 2 , y = u k int x 0 - ε 2 , y y ∈ (0, y 0 ), U k 1 2 , y = u k ext x 0 + ε 2 , y y ∈ (0, y 0 ), σ 0 ∂ X U k - 1 2 , y = σ int ∂ n u k-4 int x 0 - ε 2 , y y ∈ (0, y 0 ), σ 0 ∂ X U k 1 2 , y = σ ext ∂ n u k-4 ext x 0 + ε 2 , y y ∈ (0, y 0 ), (10)
and the following boundary conditions

               u k (0, y) = u k (L, y) = 0 y ∈ (0, y 0 ), u k (x, 0) = u k (x, y 0 ) = 0 x ∈ 0, x 0 - ε 2 ∪ x 0 + ε 2 , L , U k (X, 0) = U k (X, y 0 ) = 0 X ∈ - 1 2 , 1 2 , (11)
where δ k 0 represents the Kronecker symbol. Besides, we will also need the following equation obtained by applying the fundamental theorem of calculus for a smooth function U k ,

1 2 -1 2 ∂ 2 X U k (X, y) dX = ∂ X U k 1 2 , y -∂ X U k - 1 2 , y . (12)
If we apply the third equation of (9) on the left hand side and the third and fourth equations of (10) on the right-hand side we obtain the following compatibility condition

- 1 2 -1 2 ∂ 2 y U k-2 (X, y) dX = 1 σ 0 σ∂ n u k-4 Γ ε (y). (13)
In these equations, we adopt the convention that the terms with negative indices are equal to 0. Employing these equations ((9) -( 13)) we can deduce the elementary problems satised outside and inside the layer for any k ∈ N. For that purpose, we employ the following algorithm composed of three steps.

Algorithm for the determination of the coecients

We assume that the rst terms of the expansion (8) up to the order ε k-1 have already been calculated, and we derive the equations for the k-th term. The rst two steps consist in determining U k and the third step consist in determining u k int and u k ext .

First step:

We begin by selecting the third equation from [START_REF] Chen | New post-processing method for interpretation of through casing resistivity (TCR) measurements[END_REF], along with third and fourth equations from [START_REF] Chun | High-order accurate thin layer approximations for time-domain electromagnetics. Part II: transmission layers[END_REF], and we build the following dierential problem in the variable X for U k (the variable y plays the role of a parameter)

Inria                      ∂ 2 X U k (X, y) = -∂ 2 y U k-2 (X, y) X ∈ - 1 2 , 1 2 , σ 0 ∂ X U k - 1 2 , y = σ int ∂ n u k-4 int x 0 - ε 2 , y , σ 0 ∂ X U k 1 2 , y = σ ext ∂ n u k-4 ext x 0 + ε 2 , y . (14)
There exists a solution U k of ( 14) provided the compatibility condition ( 13) is satised. We deduce the expression of U k up to a function in the variable y, ϕ k 0 (y). The function U k has the following form

U k (X, y) = V k (X, y) + ϕ k 0 (y),
where V k represents the part of U k that can be determined at this step and has the following form (see Proposition 1)

V k (X, y) =    0 if k = 0, 1, 2, 3, ϕ k k-2 (y)X k-2 + ϕ k k-3 (y)X k-3 + . . . + ϕ k 1 (y)X if k > 3.
The function ϕ k 0 represents the part of U k that is determined at the second step.

Second step:

We employ the compatibility condition (13) (at rank k + 2), along with third equation of (11) to write the following dierential problem in the variable y for the function ϕ k 0 , involved into the expression of U k .

               d 2 dy 2 ϕ k 0 (y) = - 1 σ 0 σ∂ n u k-2 Γ ε (y) - 1 2 -1 2 ∂ 2 y V k (X, y) dX y ∈ (0, y 0 ), ϕ k 0 (0) = 0, ϕ k 0 (y 0 ) = 0. (15)
Solving this dierential equation we obtain the function ϕ k 0 and thus the complete expression of U k .

Third step:

Now we can nally derive the equations outside the layer by employing rst and second equations of (9), rst and second equations of [START_REF] Chun | High-order accurate thin layer approximations for time-domain electromagnetics. Part II: transmission layers[END_REF], and rst and second equations of [START_REF] Chun | High-order accurate thin layer approximations for timedomain electromagnetics. Part I: General metal backed coatings[END_REF]. We infer that u k int and u k ext are dened independently in the two subdomains Ω ε int and Ω ε ext by the following dierential problems

             σ int ∆u k int = f int δ k 0 in Ω ε int , u k int x 0 - ε 2 , y = U k - 1 2 , y , u k int = 0 on ∂Ω ∩ ∂Ω ε int .              σ ext ∆u k ext = f ext δ k 0 in Ω ε ext , u k ext x 0 + ε 2 , y = U k 1 2 , y , u k ext = 0 on ∂Ω ∩ ∂Ω ε ext . (16)
Subsequently, it is convenient to dene u k , for k ∈ N, by

u k =    u k int in Ω ε int , u k ext in Ω ε ext .
We will now employ this algorithm to determine the rst terms of the expansion.

First terms of the asymptotics

Case k=0

We consider Problem [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for

U 0                      ∂ 2 X U 0 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 0 - 1 2 , y = 0, ∂ X U 0 1 2 , y = 0.
We deduce that the solution to this equation has the form U 0 (X, y) = ϕ 0 0 (y). Then we employ (15) and we build the following problem for ϕ 0 0

             d 2 dy 2 ϕ 0 0 (y) = 0 y ∈ (0, y 0 ), ϕ 0 0 (0) = 0, ϕ 0 0 (y 0 ) = 0.
We deduce that ϕ 0 0 (y) = 0 and thus U 0 (X, y) = 0. Finally, employing (16), we obtain that the limit solution u 0 satises homogeneous Dirichlet boundary conditions on Γ ε int and Γ ε ext . Thus, we write the problem satised by u 0 as

Inria    σ int ∆u 0 int = f int in Ω ε int , u 0 int = 0 on ∂Ω ε int .    σ ext ∆u 0 ext = f ext in Ω ε ext , u 0 ext = 0 on ∂Ω ε ext . (17)

Case k=1

We consider Problem [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for

U 1                      ∂ 2 X U 1 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 1 - 1 2 , y = 0, ∂ X U 1 1 2 , y = 0.
We deduce that the solution to this equation has the form U 1 (X, y) = ϕ 1 0 (y). Then we employ (15) and we build the following problem for ϕ

1 0              d 2 dy 2 ϕ 1 0 (y) = 0 y ∈ (0, y 0 ), ϕ 1 0 (0) = 0, ϕ 1 0 (y 0 ) = 0.
We deduce that ϕ 1 0 (y) = 0 and thus U 1 (X, y) = 0. Finally, employing [START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF] we can write the problem satised by u 1 outside the layer as two uncoupled problems

   ∆u 1 int = 0 in Ω ε int , u 1 int = 0 on ∂Ω ε int .    ∆u 1 ext = 0 in Ω ε ext , u 1 ext = 0 on ∂Ω ε ext . (18)
We deduce that u 1 ≡ 0.

Case k=2

We consider Problem [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for

U 2                      ∂ 2 X U 2 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 2 - 1 2 , y = 0, ∂ X U 2 1 2 , y = 0.
We deduce that the solution to this equation has the form U 2 (X, y) = ϕ 2 0 (y). Then we employ (15) and we build the following problem for

ϕ 2 0              d 2 dy 2 ϕ 2 0 (y) = - 1 σ 0 σ∂ n u 0 (y) y ∈ (0, y 0 ), ϕ 2 0 (0) = 0, ϕ 2 0 (y 0 ) = 0.
We deduce that ϕ 2 0 (y) and thus U 2 (X, y) have the following form.

U 2 (X, y) = ϕ 2 0 (y) = - 1 σ 0 y 0 (y -t) σ∂ n u 0 Γ ε (t) dt + y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 Γ ε (t) dt. (19)
We assume the integrals in the expression of U 2 make sense and we make the same assumption for the rest of integrals that appear in this section. Finally, employing [START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF] we can write the problem satised outside the layer by u 2 as two uncoupled problems

           ∆u 2 int = 0 in Ω ε int , u 2 int x 0 - ε 2 , y = ϕ 2 0 (y), u 2 int = 0 on ∂Ω ∩ ∂Ω ε int .            ∆u 2 ext = 0 in Ω ε ext , u 2 ext x 0 + ε 2 , y = ϕ 2 0 (y), u 2 ext = 0 on ∂Ω ∩ ∂Ω ε ext . (20)

Case k=3

We consider Problem [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for

U 3 Inria                      ∂ 2 X U 3 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 3 - 1 2 , y = 0, ∂ X U 3 1 2 , y = 0.
We deduce that the solution to this equation has the form U 3 (X, y) = ϕ 3 0 (y). Then we employ (15) and we build the following problem for

ϕ 3 0              d 2 dy 2 ϕ 3 0 (y) = 0 y ∈ (0, y 0 ), ϕ 3 0 (0) = 0, ϕ 3 0 (y 0 ) = 0.
We deduce that ϕ 3 0 (y) = 0 and thus U 3 (X, y) = 0. Finally, employing ( 16) we can write the problem satised outside the layer by u 3 as two uncoupled problems

   ∆u 3 int = 0 in Ω ε int , u 3 int = 0 on ∂Ω ε int .    ∆u 3 ext = 0 in Ω ε ext , u 3 ext = 0 on ∂Ω ε ext . (21)
We deduce that u 3 ≡ 0.

Recapitulation of the asymptotic expansion

Proposition 1. The asymptotic expansion [START_REF] Caloz | Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism[END_REF], has the following form

               u int (x, y) = u 0 int (x, y) + ε 2 u 2 int (x, y) + O ε 4 in Ω ε int , u ext (x, y) = u 0 ext (x, y) + ε 2 u 2 ext (x, y) + O ε 4 in Ω ε ext , U (X, y) = ε 2 ϕ 2 0 (y) + O ε 4 in - 1 2 , 1 2 × (0, y 0 ),
where the functions ϕ 2 0 , u 0 and u 2 are dened by Equations ( 19), [START_REF] Durué | Thin layer models for electromagnetism[END_REF] and [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF] respectively. Besides, for k ∈ N, the solution U k to Equation [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] has the following form

U k (X, y) =          0 if k = 0 or k odd, k-2 j=0 ϕ k j (y)X j if k even,
and the solution u k = (u k int , u k ext ) to Problem [START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF] satises

u k int ≡ u k ext ≡ 0, if k odd.
Proof. We perform the proof by induction on k. For k = 0, 1, 2, 3, we have already calculated the expressions of u k and U k in the previous section. Now let us assume that for all even number i ∈ N, such that i < k, the function U i has the form

U i (X, y) = ϕ i i-2 (y)X i-2 + ϕ i i-3 (y)X i-3 + . . . + ϕ i 1 (y)X + ϕ i 0 (y),
We begin by considering Problem [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for any even number k ≥ 4. Solving this problem we obtain a solution of the form

U k (X, y) = ϕ k k-2 (y)X k-2 + ϕ k k-3 (y)X k-3 + . . . + ϕ k 1 (y)X + ϕ k 0 (y),
where k ≥ 4 and

                     ϕ k 1 (y) = 1 σ 0 σ∂ n u k-4 (y), ϕ k 2 (y) = 1 2 σ 0 σ∂ n u k-4 (y), ϕ k k-j (y) = -d 2 dy 2 ϕ k-2 k-j-2 (y) (k -j)(k -j -1) j = 2, . . . , k -2.
We denote by V k the following function

V k (X, y) = ϕ k k-2 (y)X k-2 + ϕ k k-3 (y)X k-3 + . . . + ϕ k 1 (y)X.
This expression corresponds to the function V k we have dened at the rst step of the algorithm. The only thing left to prove is that if k is an odd number, U k ≡ 0 and u k int ≡ u k ext ≡ 0. We assume that for all odd number j ∈ N, such that j < k, U j ≡ 0 and u j int ≡ u j ext ≡ 0. Employing Equation ( 14), we have the following problem for

U k                      ∂ 2 X U k (X, y) = -∂ 2 y U k-2 (X, y) X ∈ - 1 2 , 1 2 , σ 0 ∂ X U k - 1 2 , y = σ int ∂ n u k-4 int x 0 - ε 2 , y , σ 0 ∂ X U k 1 2 , y = σ ext ∂ n u k-4 ext x 0 + ε 2 , y .
Thanks to the inductive assumptions we know that U k-2 ≡ 0 and u k-4 int ≡ u k-4 ext ≡ 0. Thus, we deduce that U k has the following form

U k (X, y) = ϕ k 0 (y).

Now we employ Equation (15) to build the following problem

Inria              d 2 dy 2 ϕ k 0 (y) = - 1 σ 0 σ∂ n u k-2 Γ ε (y) y ∈ (0, y 0 ), ϕ k 0 (0) = 0, ϕ k 0 (y 0 ) = 0.
Again, thanks to the inductive assumptions we know that u k-2 ≡ 0. Thus, we deduce that ϕ k 0 ≡ 0 and consequently U k ≡ 0. Now employing ( 16), we can write the problems satised by u k int and u k ext .

         ∆u k int = 0 in Ω ε int , u k int = 0 on Γ ε int , u k int = 0 on ∂Ω ∩ ∂Ω ε int .          ∆u k ext = 0 in Ω ε ext , u k ext = 0 on Γ ε ext , u k ext = 0 on ∂Ω ∩ ∂Ω ε ext . We deduce that u k ≡ 0.

Equivalent models

Now that we know the expressions for the rst terms of the expansion, we truncate the series for a given k ∈ N and we identify a simpler problem satised by

u (k) = u 0 + εu 1 + . . . + ε k u k in Ω ε int ∪ Ω ε
ext up to a residual term of order ε k+1 . We neglect the residual term of order ε k+1 to obtain an approximate model satised by the function u [k] . For the sake of simplicity of the notation we will employ the following notation for the domain.

Notation 2. We denote by Ω ε the domain

Ω ε = Ω ε int ∪ Ω ε ext ,
where Ω ε int and Ω ε ext are the domains dened in Section 1.

In the following we dene what the order of convergence for an asymptotic model is. Here, we formally derive two approximate models of order 2 and order 4 respectively. These orders of convergence have still to be proven, we refer the reader to Section 4 for more details concerning these convergence proofs. Denition 2. Let u [k] be the solution to an asymptotic model, and let u be the solution to the reference problem. We say that the asymptotic model is of order k + 1, if there exists a constant C independent of ε, such that the following relation is satised

u -u [k] 1,Ω ε ≤ Cε k+1 ,
for ε suciently small. RR n°8998

Second-order model

For deriving the second-order model, we truncate the series from the second term and we dene u (1) as

u (1) = u 0 + εu 1 = u 0 in Ω ε int ∪ Ω ε ext (see Proposition 1).
From [START_REF] Durué | Thin layer models for electromagnetism[END_REF], we can deduce that u (1) solves the following uncoupled problem

   σ int ∆u (1) int = f int in Ω ε int , u (1)
int = 0 on ∂Ω ε int .    σ ext ∆u (1) ext = f ext in Ω ε ext , u (1)
ext = 0 on ∂Ω ε ext . (22)
In this case, we have u [1] = u (1) as u (1) does not depend on ε. We infer a second order model satised by u [1] solution to Problem [START_REF] Haddar | Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers[END_REF].

Fourth-order model

For deriving the fourth-order model, we truncate the series from the fourth term and we dene u (3) as

u (3) = u 0 + εu 1 + ε 2 u 2 + ε 3 u 3 = u 0 + ε 2 u 2 in Ω ε int ∪ Ω ε ext (see Proposition 1).
From ( 17), ( 18), ( 20) and ( 21), we can deduce that u (3) satises the following equations

                               σ int ∆u (3) int = f int in Ω ε int , σ ext ∆u (3) ext = f ext in Ω ε ext , u (3)
Γ ε = 0, d 2 dy 2 u (3) Γ ε = -ε 2 1 σ 0 σ∂ n u 0 Γ ε , u (3) = 0 on ∂Ω ∩ ∂Ω ε .
Then, we employ the expression u 0 = u (3) -ε 2 u 2 to rewrite the right-hand side of the second transmission condition

-ε 2 1 σ 0 σ∂ n u 0 Γ ε = -ε 2 1 σ 0 σ∂ n u (3) Γ ε + ε 4 1 σ 0 σ∂ n u 2 Γ ε = -ε 2 1 σ 0 σ∂ n u (3) Γ ε + O(ε 4 ).
Now we deduce that u (3) satises the following equations

Inria                                σ int ∆u (3) int = f int in Ω ε int , σ ext ∆u (3) ext = f ext in Ω ε ext , u (3) Γ ε = 0, d 2 dy 2 u (3) Γ ε = -ε 2 1 σ 0 σ∂ n u (3) Γ ε + O(ε 4 ), u (3) = 0 on ∂Ω ∩ ∂Ω ε .
We dene as u [3] the function we obtain when truncating the solution at the fourth element of the expansion and neglecting the terms of order 4 or higher in ε. Then, u [3] satises the following equations

                               σ int ∆u [3] int = f int in Ω ε int , σ ext ∆u [3] ext = f ext in Ω ε ext , u [3] Γ ε = 0, σ∂ n u [3] Γ ε = - σ 0 ε 2 d 2 dy 2 u [3] Γ ε , u [3] = 0 on ∂Ω ∩ ∂Ω ε . (23)

Classical conditions and comparison with equivalent conditions

In this section we show the results we obtain when we no longer consider the conductivity in the thin layer to be dependent on its thickness to remark the dierent results we obtain with our approach compared to this one. The model problem remains the same, we consider Equations (2) set in the domain showed in Figure 1, but know we consider a conductivity of the following form

σ =          σ int in Ω ε int , σ lay in Ω ε lay , σ ext in Ω ε ext ,
where σ lay is just a constant and not dependent on ε any more. With this conguration, the model problem we consider writes as follows

                                           σ int ∆u int = f int in Ω ε int , σ ext ∆u ext = f ext in Ω ε ext , ∆u lay = 0 in Ω ε lay , u int = u lay on Γ ε int , u lay = u ext on Γ ε ext , σ int ∂ n u int = σ lay ∂ n u lay on Γ ε int , σ lay ∂ n u lay = σ ext ∂ n u ext on Γ ε ext , u = 0 on ∂Ω. (24)
Considering this model, and applying the same asymptotic method developed in the previous section to derive approximate models, we obtain a rst-order model and a third-order model. The expression for these models are the following.

First-order model

                             σ int ∆u [0] int = f int in Ω ε int , σ ext ∆u [0] ext = f ext in Ω ε ext , u [0] Γ ε = 0, σ∂ n u [0] Γ ε = 0, u [0] = 0 on ∂Ω ∩ ∂Ω ε . (25)
Third-order model

                               σ int ∆u [2] int = f int in Ω ε int , σ ext ∆u [2] ext = f ext in Ω ε ext , u [2] Γ ε = ε σ lay σ∂ n u [2] Γ ε , σ∂ n u [2] Γ ε = -εσ lay d 2 dy 2 u [2] Γ ε , u [2] = 0 on ∂Ω ∩ ∂Ω ε . ( 26
)
We notice that these models are dierent from ( 22) and [START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF]. A main dierence comparing with our approach (i.e. when σ lay = σ 0 ε -3 ) comes from the fact that now the lower order model ( 25) is coupled, whereas model ( 22) is governed by two independent problems. Moreover, the lower order model (25) has order 1, whereas model ( 22) has order 2. In the same way, the higher order model ( 26) is of order 3, whereas model ( 23) is of order 4. Second class of ITCs

The objective of this section is the derivation of asymptotic models for the reference Problem [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF]. The main dierence with the rst class of ITCs is that now we employ some formal Taylor expansions to write the terms of the expansion across an articial interface Γ situated in the middle of the thin layer. The resulting asymptotic models will be dened in the domain depicted in Figure 2b. In this section we perform an expansion of the solution in power series of ε, following the same way as in Section 2.1. Then, by truncating this series and neglecting higher order terms in ε, we derive approximate models composed by equivalent transmission conditions in Section 3.2. Finally in Section 3.3 we present a technique for solving a stability problem related with one of the derived asymptotic models. The last section, Section 3.4, shows a comparison of the models derived here with more standard models.

Construction of a multiscale expansion

First of all, we dene the jump and mean value of a function across the interface Γ, in the same way we have done with the jump and the mean value across the thin layer in Denition 1. Denition 3. Let u be a function dened over Ω. We dene its jump and mean value across the interface Γ as

Ω ε int Ω ε ext Ω ε lay Γ Γ ε int Γ ε ext x0 L y0 x y (
     [u] Γ = u ext | Γ -u int | Γ , {u} Γ = 1 2 (u ext | Γ + u int | Γ ) .
Now we derive the asymptotic expansion. To begin with, we perform an Ansatz in the form of power series of ε for the solution to Problem [START_REF] Caloz | On the inuence of the geometry on skin eect in electromagnetism[END_REF]. We look for solutions

RR n°8998                        u int (x, y) ≈ k≥0 ε k u k int (x, y) in Ω ε int , u ext (x, y) ≈ k≥0 ε k u k ext (x, y) in Ω ε ext , U (X, y) ≈ k≥0 ε k U k (X, y) in - 1 2 , 1 2 × (0, y 0 ) , (27)
where the functions u k int k∈N and u k ext k∈N are now dened in ε-independent domains, contrary to the rst approach. We emphasize that the sequence u k int k∈N (respectively u k ext k∈N ) is dened in Ω int (respectively Ω ext ) even if its associated series does not approach u in the thin layer. We assume that for k ∈ N, the terms u k int and u k ext are as regular as necessary, see [START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF]. Then, we perform a formal Taylor expansion of the terms u k int Γ ε int and u k ext Γ ε ext of the series along the direction normal to the thin layer, the variable x in this case, in order to write the transmission conditions across the interface Γ. The formal Taylor expansion writes as follows

             u k int x 0 - ε 2 , y = i≥0 ε i (-1) i 2 i i! ∂ i n u k int (x 0 , y) , u k ext x 0 + ε 2 , y = i≥0 ε i 1 2 i i! ∂ i n u k ext (x 0 , y) .
We also perform a formal Taylor expansion of the same form for the derivatives

∂ n u k int Γ ε int and ∂ n u k ext Γ ε ext              ∂ n u k int x 0 - ε 2 , y = i≥0 (-1) i 2 i i! ∂ i+1 n u k int (x 0 , y) , ∂ n u k ext x 0 + ε 2 , y = i≥0 ε i 1 2 i i! ∂ i+1 n u k ext (x 0 , y) .
Employing these formal Taylor expansions and the Ansatz [START_REF] Kaufman | A transmission-line model for electrical logging through casing[END_REF] we develop the terms u int Γ ε int and u ext Γ ε ext in the following way

u int Γ ε int = u int x 0 - ε 2 , y = k≥0 ε k u k int x 0 - ε 2 , y = k≥0 ε k i≥0 ε i (-1) i 2 i i! ∂ i n u k int (x 0 , y) = k≥0 ε k k i=0 (-1) i 2 i i! ∂ i n u k-i int (x 0 , y), u ext Γ ε ext = u ext x 0 + ε 2 , y = k≥0 ε k u k ext x 0 + ε 2 , y = k≥0 ε k i≥0 ε i 1 2 i i! ∂ i n u k ext (x 0 , y) = k≥0 ε k k i=0 1 2 i i! ∂ i n u k-i ext (x 0 , y), (28)
Inria and the terms

∂ n u k int Γ ε int , ∂ n u k ext Γ ε ext
in the following way

∂ n u int Γ ε int = ∂ n u int x 0 - ε 2 , y = k≥0 ε k ∂ n u k int x 0 - ε 2 , y = k≥0 ε k i≥0 ε i (-1) i 2 i i! ∂ i+1 n u k int (x 0 , y) = k≥0 ε k k i=0 (-1) i 2 i i! ∂ i+1 n u k-i int (x 0 , y), (29)
∂ n u ext Γ ε ext = ∂ n u ext x 0 + ε 2 , y = k≥0 ε k ∂ n u k ext x 0 + ε 2 , y = k≥0 ε k i≥0 ε i 1 2 i i! ∂ i+1 n u k ext (x 0 , y) = k≥0 ε k k i=0 1 2 i i! ∂ i+1 n u k-i ext (x 0 , y).

Equations for the coecients of the electric potential

Substituting the Ansatz [START_REF] Kaufman | A transmission-line model for electrical logging through casing[END_REF] and the identities ( 28), [START_REF] Pardo | PML enhanced with a self-adaptive goal-oriented hp-nite element method: Simulation of through-casing borehole resistivity measurements[END_REF] in the Equations ( 7) and collecting the terms with the same powers in ε, for every k ∈ N we obtain the following set of equations

             σ int ∆u k int (x, y) = f int (x, y)δ k 0 in Ω int , σ ext ∆u k ext (x, y) = f ext (x, y)δ k 0 in Ω ext , ∂ 2 X U k (X, y) + ∂ 2 y U k-2 (X, y) = 0 in - 1 2 , 1 2 × (0, y 0 ) , (30)
along with the following transmission conditions

                                         k i=0 (-1) i 2 i i! ∂ i n u k-i int (x 0 , y) = U k -1 2 , y y ∈ (0, y 0 ), k i=0 1 2 i i! ∂ i n u k-i ext (x 0 , y) = U k 1 2 , y y ∈ (0, y 0 ), σ int k-4 i=0 (-1) i 2 i i! ∂ i+1 n u k-4-i int (x 0 , y) = σ 0 ∂ X U k -1 2 , y y ∈ (0, y 0 ), σ ext k-4 i=0 1 2 i i! ∂ i+1 n u k-4-i ext (x 0 , y) = σ 0 ∂ X U k 1 2 , y y ∈ (0, y 0 ), (31)
and the following boundary conditions

             u k (0, y) = u k (L, y) = 0 y ∈ (0, y 0 ) , u k (x, 0) = u k (x, y 0 ) = 0 x ∈ (0, x 0 ) ∪ (x 0 , L) , U k (X, 0) = U k (X, y 0 ) = 0 X ∈ - 1 2 , 1 2 , (32)
where δ k 0 represents the Kronecker symbol. Besides, we will also employ the following compatibility condition obtained by applying the fundamental theorem of calculus [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF] for a smooth function U k+2 , along with the third equation of [START_REF] Pardo | Two-Dimensional High-Accuracy Simulation of Resistivity Log ging-While-Drilling (LWD) Measurements Using a Self-Adaptive Goal-Oriented hp Finite Element Method[END_REF] and the third and fourth equations of ( 32)

- 1 2 -1 2 ∂ 2 y U k (X, y) dX = 1 σ 0 k-2 i=0 σ ext 2 i i! ∂ i+1 n u k-2-i ext (x 0 , y) + (-1) i+1 σ int 2 i i! ∂ i+1 n u k-2-i int (x 0 , y) (33)
For these equations, we adopt the convention that the terms with negative indices are equal to 0. Employing these equations (( 30) -( 33)) we can determine the elementary problems satised outside and inside the layer for any k ∈ N. For that purpose, we use the following algorithm composed of three steps.

Algorithm for the determination of the coecients

We assume that the rst terms of the expansion [START_REF] Kaufman | A transmission-line model for electrical logging through casing[END_REF] up to order ε k-1 have already been calculated and we calculate the equations for the k-th term. The rst two steps consist in xing U k and the third step consist in determining u k int and u k ext .

First step:

We begin by selecting the third equation from [START_REF] Pardo | Two-Dimensional High-Accuracy Simulation of Resistivity Log ging-While-Drilling (LWD) Measurements Using a Self-Adaptive Goal-Oriented hp Finite Element Method[END_REF], along with third and fourth equations from [START_REF] Pardo | A self-adaptive goaloriented hp-nite element method with electromagnetic applications[END_REF], and we build the following dierential problem in the variable X for U k (the variable y plays the role of a parameter)

                         ∂ 2 X U k (X, y) = -∂ 2 y U k-2 (X, y) X ∈ - 1 2 , 1 2 , σ 0 ∂ X U k -1 2 , y = σ int k-4 i=0 (-1) i 2 i i! ∂ i+1 n u k-4-i int (x 0 , y) , σ 0 ∂ X U k 1 2 , y = σ ext k-4 i=0 1 2 i i! ∂ i+1 n u k-4-i ext (x 0 , y) . (34)
There exists a solution U k to (34) provided the compatibility condition (33) is satised. We deduce the expression of U k up to a function in the variable y, ψ k 0 (y). The function U k has the following form

U k (X, y) = V k (X, y) + ψ k 0 (y),
Inria where V k represents the part of U k that can be determined at this step and has the form (see Proposition 2)

V k (X, y) =    0 if k = 0, 1, 2, 3,
ψ k k-2 (y)X k-2 + ψ k k-3 (y)X k-3 + . . . + ψ k 1 (y)X if k > 3.
The function ψ k 0 represents the part of U k that is determined at the second step.

Second step:

We involve the compatibility condition [START_REF] Pardo | Simulation of multifrequency borehole resistivity measurements through metal casing using a goal-oriented hp nite-element method[END_REF], along with third equation of [START_REF] Pardo | A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: electrostatics[END_REF] to write the following dierential problem in the variable y for the function ψ k 0 , present in the expression of

U k .              d 2 dy 2 ψ k 0 (y) = h k (y) y ∈ (0, y 0 ), ψ k 0 (0) = 0, ψ k 0 (y 0 ) = 0, (35)
where

h k (y) = - 1 σ 0 k-2 i=0 σ ext 2 i i! ∂ i+1 n u k-2-i ext (x 0 , y) + (-1) i+1 σ int 2 i i! ∂ i+1 n u k-2-i int (x 0 , y) - 1 2 -1 2 ∂ 2 y V k (X, y) dX.
Solving this dierential equation we obtain the function ψ k 0 and thus the complete expression of U k .

Third step:

Now we can nally derive the equations outside the layer by employing rst and second equations of [START_REF] Pardo | Two-Dimensional High-Accuracy Simulation of Resistivity Log ging-While-Drilling (LWD) Measurements Using a Self-Adaptive Goal-Oriented hp Finite Element Method[END_REF], rst and second equations of [START_REF] Pardo | A self-adaptive goaloriented hp-nite element method with electromagnetic applications[END_REF], and rst and second equations of [START_REF] Pardo | A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: electrostatics[END_REF]. We infer that u k int and u k ext are dened independently in the two subdomains Ω int and

Ω ext            σ int ∆u k int = f int δ k 0 in Ω int , u k int (x 0 , y) = U k - 1 2 , y - k i=1 (-1) i 2 i i! ∂ i n u k-i int (x 0 , y) , u k int = 0 on ∂Ω ∩ ∂Ω int . (36)
           σ ext ∆u k ext = f ext δ k 0 in Ω ext , u k ext (x 0 , y) = U k 1 2 , y - k i=1 1 2 i i! ∂ i n u k-i ext (x 0 , y) , u k ext = 0 on ∂Ω ∩ ∂Ω ext .
Now this algorithm is used to determine the rst terms of the expansion.

First terms of the asymptotics

Case k=0

We consider Problem [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] for

U 0                      ∂ 2 X U 0 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 0 - 1 2 , y = 0, ∂ X U 0 1 2 , y = 0.
We deduce that the solution to this equation has the form U 0 (X, y) = ψ 0 0 (y). Then using [START_REF] Pardo | Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp nite-element method. Part II: through-casing resistivity instruments[END_REF] we build the following problem for

ψ 0 0              d 2 dy 2 ψ 0 0 (y) = 0 y ∈ (0, y 0 ), ψ 0 0 (0) = 0, ψ 0 0 (y 0 ) = 0.
We deduce that ψ 0 0 (y) = 0 and thus U 0 (X, y) = 0. Finally, employing [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF] we obtain that the limit solution u 0 satises homogeneous Dirichlet boundary conditions on Γ. Thus, the problem satised by u 0 reads as

Inria    ∆u 0 int = f int in Ω int , u 0 int = 0 on ∂Ω int .    ∆u 0 ext = f ext in Ω ext , u 0 ext = 0 on ∂Ω ext . (37)

Case k=1

We consider Problem [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] for

U 1                      ∂ 2 X U 1 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 1 - 1 2 , y = 0, ∂ X U 1 1 2 , y = 0.
We deduce that the solution to this equation has the form U 1 (X, y) = ψ 1 0 (y). Then from (35) we deduce the following problem for

ψ 1 0              d 2 dy 2 ψ 1 0 (y) = 0 y ∈ (0, y 0 ), ψ 1 0 (0) = 0, ψ 1 0 (y 0 ) = 0.
We thus have that ψ 1 0 (y) = 0 and U 1 (X, y) = 0. Finally, employing [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF] we can write the problem satised outside the layer by u 1 as two uncoupled problems

           ∆u 1 int = 0 in Ω int , u 1 int = 1 2 ∂ n u 0 int on Γ, u 1 int = 0 on ∂Ω ∩ ∂Ω int .            ∆u 1 ext = 0 in Ω ext , u 1 ext = - 1 2 ∂ n u 0 ext on Γ, u 1 ext = 0 on ∂Ω ∩ ∂Ω ext . (38)

Case k=2

We consider Problem [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] for

U 2                      ∂ 2 X U 2 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 2 - 1 2 , y = 0, ∂ X U 2 1 2 , y = 0.
We deduce that the solution to this equation has the form U 2 (X, y) = ψ (y). Then according to [START_REF] Pardo | Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp nite-element method. Part II: through-casing resistivity instruments[END_REF] we can build the following problem for

ψ 2 0              d 2 dy 2 ψ 2 0 (y) = - 1 σ 0 σ ext ∂ n u 0 (y) y ∈ (0, y 0 ), ψ 2 0 (0) = 0, ψ 2
0 (y 0 ) = 0. We deduce that ψ 2 0 has the following form

ψ 2 0 (y) = -1 σ 0 y 0 (y -t) σ∂ n u 0 (t) dt + y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 (t) dt. (39)
Finally, [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF] implies that the problem satised outside the layer by u 2 is composed of two uncoupled problems

         ∆u 2 int = 0 in Ω int , u 2 int (x 0 , y) = ψ 2 0 (y) y ∈ (0, y 0 ), u 2 int = 0 on ∂Ω ∩ ∂Ω int .          ∆u 2 ext = 0 in Ω ext , u 2 ext (x 0 , y) = ψ 2 0 (y) y ∈ (0, y 0 ), u 2 ext = 0 on ∂Ω ∩ ∂Ω ext . (40)

Case k=3

We consider Problem [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] for

U 3                      ∂ 2 X U 3 (X, y) = 0 X ∈ -1 2 , 1 2 , ∂ X U 3 - 1 2 , y = 0, ∂ X U 3 1 2 , y = 0.
We deduce that the solution to this equation has the form U 3 (X, y) = ψ (y). Then we employ [START_REF] Pardo | Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp nite-element method. Part II: through-casing resistivity instruments[END_REF] and we build the following problem for ψ 3 0

Inria              d 2 dy 2 ψ 3 0 (y) = - 1 σ 0 σ ext ∂ n u 1 (y) - 1 σ 0 σ ext ∂ n u 0 (y) y ∈ (0, y 0 ), ψ 3 0 (0) = 0, ψ 3 0 (y 0 ) = 0.
We deduce that ψ 3 0 has the following form

ψ 3 0 (y) = -1 σ 0 y 0 (y -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt + y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt. (41)
Finally, employing [START_REF] Pardo | Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers[END_REF] we can write the problem satised outside the layer by u 3 as two uncoupled problems

         ∆u 3 int = 0 in Ω int , u 3 int (x 0 , y) = ψ 3 0 (y) y ∈ (0, y 0 ), u 3 int = 0 on ∂Ω ∩ ∂Ω int .          ∆u 3 ext = 0 in Ω ext , u 3
ext (x 0 , y) = ψ 3 0 (y) y ∈ (0, y 0 ),

u 3 ext = 0 on ∂Ω ∩ ∂Ω ext . (42)
3.1.4 Recapitulation of the asymptotic expansion Proposition 2. The asymptotic expansion [START_REF] Kaufman | A transmission-line model for electrical logging through casing[END_REF], has the following form,

   u int (x, y) = u 0 int (x, y) + εu 1 int (x, y) + ε 2 u 2 int (x, y) + ε 3 u 3 int (x, y) + O ε 4 , u ext (x, y) = u 0 ext (x, y) + εu 1 ext (x, y) + ε 2 u 2 int (x, y) + ε 3 u 3 int (x, y) + O ε 4 ,
and the following form inside the layer, for (X, y) ∈ -

1 2 , 1 2
× (0, y 0 ) U (X, y) = ε 2 ψ 2 0 (y) + ε 3 ψ 3 0 (y) + O ε 4 ,
where the functions u 0 , u 1 , u 2 and u 3 are dened by Equations ( 37), [START_REF] Péron | Mathematical modeling of electromagnetic phenomena in high contrast media[END_REF], [START_REF] Péron | Equivalent boundary conditions for an elasto-acoustic problem set in a domain with a thin layer[END_REF] and [START_REF] Péron | Equivalent transmission conditions for the timeharmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet[END_REF] respectively, and the functions ψ 2 0 and ψ 3 0 are dened by Equations [START_REF] Péron | Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer[END_REF] and [START_REF] Péron | Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer[END_REF] respectively. Besides, for k ∈ N, the solution U k to Equation [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] has the following form

U k (X, y) =                        0 k = 0, 1, k-2 j=0 ψ k j (y)X j k = 2l + 2, l ∈ N, k-3 j=0 ψ k j (y)X j k = 2l + 3, l ∈ N.
Proof. We perform the proof by induction on k. For k = 0, 1, 2, 3 we have already calculated the expressions of u k and U k in the previous section. Now let us assume that for any even number i ∈ N, such that i < k, the function U i has the form

U i (X, y) = ψ i i-2 (y)X i-2 + ψ i i-3 (y)X i-3 + . . . + ψ i 1 (y)X + ψ i 0 (y),
We begin by considering Problem [START_REF] Pardo | Feasibility study for 2D frequencydependent electromagnetic sensing through casing[END_REF] for any even number k. Solving this problem we obtain a solution of the form

U k (X, y) = ψ k k-2 (y)X k-2 + ψ k k-3 (y)X k-3 + . . . + ψ k 1 (y)X + ψ k 0 (y), where k ≥ 2 and                            ψ k 1 (y) = 1 σ 0 k-4 i=0 σ ext 2 i i! ∂ i+1 n u k-4-i ext (x 0 , y) + (-1) i σ int 2 i i! ∂ i+1 n u k-4-i int (x 0 , y) , ψ k 2 (y) = 1 2 σ 0 k-4 i=0 σ ext 2 i i! ∂ i+1 n u k-4-i ext (x 0 , y) + (-1) i+1 σ int 2 i i! ∂ i+1 n u k-4-i int (x 0 , y) , ψ k k-j (y) = -d 2 dy 2 ψ k-2 k-j-2 (y) (k -j)(k -j -1) j = 2, . . . , k -2.
We denote by V k the following function

V k (X, y) = ψ k k-2 (y)X k-2 + ψ k k-3 (y)X k-3 + . . . + ψ k 1 (y)X.
This expression corresponds to the function V k we have dened in the rst step of the algorithm. A similar argument can be involved when k is an odd number.

Equivalent models

Now that we know the expressions for the rst terms of the expansion, we truncate the series and we identify a simpler problem satised by

u (k) = u 0 + εu 1 + . . . + ε k u k in Ω int ∪ Ω ext
up to a residual term of order ε k+1 . We neglect the residual term of order ε k+1 to obtain an approximate model satised by the function u [k] . Here, we formally derive several approximate models of order 1, order 2, order 3 and order 4 respectively. Inria

First-order model

For deriving the rst-order model, we truncate the series from the rst term and we dene u (0) as

u (0) = u 0 in Ω int ∪ Ω ext (see Proposition 2).
From [START_REF] Pérez-Arancibia | On the Green's function for the Helmholtz operator in an impedance circular cylindrical waveguide[END_REF], we can deduce that u (0) solves the problem

   σ int ∆u (0) int = f int in Ω int , u (0) int = 0 on ∂Ω int .    σ ext ∆u (0) ext = f ext in Ω ext , u (0) ext = 0 on ∂Ω ext . (43)
In this case, we have u [0] = u (0) as u (0) does not depend on ε. We thus infer a rst-order model satised by u [0] solution to Problem (43).

Second-order model

For deriving the second-order model, we truncate the series from the second term and we dene u (1) as

u (1) = u 0 + εu 1 in Ω int ∪ Ω ext (see Proposition 2).
From [START_REF] Pérez-Arancibia | On the Green's function for the Helmholtz operator in an impedance circular cylindrical waveguide[END_REF] and [START_REF] Péron | Mathematical modeling of electromagnetic phenomena in high contrast media[END_REF] we can deduce that u (1) satises the following equations

             σ int ∆u (1) int = f int in Ω int , u (1) int
= ε 2 ∂ n u 0 int on Γ, u (1)
int = 0 on ∂Ω ∩ ∂Ω int .              σ ext ∆u (1) ext = f ext in Ω ext , u (1)
ext = -

ε 2 ∂ n u 0 ext on Γ, u (1)
ext = 0 on ∂Ω ∩ ∂Ω ext .

(

) 44
Following the same procedure as in Section 2.2 we obtain the following second-order asymptotic model for u [1]

             σ int ∆u [1] int = f int in Ω int , u [1] int = ε 2 ∂ n u [1] int on Γ, u [1] int = 0 on ∂Ω ∩ ∂Ω int .              σ ext ∆u [1] ext = f ext in Ω ext , u [1] ext = - ε 2 ∂ n u [1] ext on Γ, u [1] ext = 0 on ∂Ω ∩ ∂Ω ext . (45)

Third-order model

For deriving the third-order model, we truncate the series from the third term and we dene u (2) as

u (2) = u 0 + εu 1 + ε 2 u 2 in Ω int ∪ Ω ext (see Proposition 2).
From [START_REF] Pérez-Arancibia | On the Green's function for the Helmholtz operator in an impedance circular cylindrical waveguide[END_REF], [START_REF] Péron | Mathematical modeling of electromagnetic phenomena in high contrast media[END_REF] and [START_REF] Péron | Equivalent boundary conditions for an elasto-acoustic problem set in a domain with a thin layer[END_REF] we can deduce that u (2) satises the following equations

           σ int ∆u (2) int = f int in Ω int , u (2)
int (x 0 , y) = g 2 1 (y) y ∈ (0, y 0 ), u

int = 0 on ∂Ω ∩ ∂Ω int .

           σ ext ∆u (2) ext = f ext in Ω ext , u (2)
ext (x 0 , y) = g 2 2 (y) y ∈ (0, y 0 ), u

ext = 0 on ∂Ω ∩ ∂Ω ext . ( (2)
) 46
where g 2 1 and g 2 2 are dened as follows

                                 g 2 1 (y) = ε 2 ∂ n u 0 int (x 0 , y) - ε 2 σ 0 y 0 (y -t) σ∂ n u 0 (t) dt + ε 2 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 (t) dt, g 2 2 (y) = - ε 2 ∂ n u 0 ext (x 0 , y) - ε 2 σ 0 y 0 (y -t) σ∂ n u 0 (t) dt + ε 2 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 (t) dt.
Following the same procedure as in Section 2.2 we obtain the following third-order asymptotic model for u [2]

Inria                              σ int ∆u [2] int = f int in Ω int , σ ext ∆u [2] ext = f ext in Ω ext , T 2 1 u [2] = 0 on Γ, T 2 2 u [2] = 0 on Γ, u [2] ext = 0 on ∂Ω. ( 47
)
where T 2 1 and T 2 2 are dened as follows

         T 2 1 u [2] = u [2] Γ + ε ∂ n u [2] Γ + ε 2 8 ∂ n u [2] Γ , T 2 2 u [2] = ε 2 σ 0 σ∂ n u [2] Γ + ε 2 8 d 2 dy 2 ∂ 2 n u [2] Γ + ε 4 d 2 dy 2 ∂ n u [2] Γ + d 2 dy 2 u [2]                                                                          g (y) = ε 2 ∂ n u 0 int (x 0 , y) - ε 2 σ 0 y 0 (y -t) σ∂ n u 0 (t) dt + ε 3 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 (t) dt - ε 3 σ 0 y 0 (y -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt + ε 3 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt, g (y) = - ε 2 ∂ n u 0 ext (x 0 , y) - ε 2 σ 0 y 0 (y -t) σ∂ n u 0 (t) dt + ε 2 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 0 (t) dt - ε 3 σ 0 y 0 (y -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt + ε 3 y σ 0 y 0 y0 0 (y 0 -t) σ∂ n u 1 (t) + σ∂ n u 0 (t) dt.
Following the same procedure as in Section 2.2 we obtain the following fourth-order asymptotic model for u [3]

                             σ int ∆u [3] int = f int in Ω int , σ ext ∆u [3] ext = f ext in Ω ext , T 3 1 u [3] = 0 on Γ, T 3 2 u [3] = 0 on Γ, u [3] ext = 0 on ∂Ω. ( 49
)
where T 3 1 and T 3 2 are dened as follows

                   T 3 1 u [3] = u [3] Γ + ε ∂ n u [3] Γ + ε 2 8 ∂ n u [3] Γ + ε 3 24 ∂ 3 n u [3] Γ , T 3 2 u [3] = ε 2 σ 0 σ∂ n u [3] Γ + ε 3 σ 0 σ∂ n u [3] Γ + ε 3 96 d 2 dy 2 ∂ 3 n u [3] Γ + ε 2 8 d 2 dy 2 ∂ 2 n u [3] Γ + ε 4 d 2 dy 2 ∂ n u [3] Γ + d 2 dy 2 u [3] Γ .

Articial boundaries

If we derive the variational formulation of the second-order model ( 45), as we do later in the Section 4.3.2, we notice that we cannot prove the coerciveness of the bilinear form, dened in Inria Section 4.3.2, due to a negative term. This negative term could cause instabilities when solving the problem with the nite element method as we observe in Section 5.2. However, to overcome this problem and restore stability we are going to use a technique based on introducing some new articial boundaries, across of which we are going to rewrite the transmission conditions, see for example [START_REF] Delourme | Modèles et asymptotiques des interfaces nes et périodiques en électromagnétisme[END_REF][START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF]. We dene these new articial boundaries as follows Denition 4. We dene the articial boundaries Γ δ int and Γ δ ext as We can observe the new conguration dened by the articial boundaries in Figure 3.

Γ δ int = {(x 0 -δε, y) : δ > 0, y ∈ (0, y 0 )} , Γ δ ext = {(x 0 + δε, y) : δ > 0, y ∈ (0, y 0 )} . Ω δ int Ω δ ext Γ Γ δ int Γ δ
Remark 2. The domains Ω δ int and Ω δ ext , and the boundaries Γ δ int and Γ δ ext , all depend on ε, but we do not include it in the notation for the sake of simplicity.

We apply a formal Taylor expansion on the variable normal to the thin layer, x in this case, in order to write the transmission conditions across the articial boundaries.

                     u ext (x 0 , y) = u ext (x 0 + δε, y) -δε∂ n u ext (x 0 + δε, y) + O(ε 2 ), u int (x 0 , y) = u int (x 0 -δε, y) + δε∂ n u int (x 0 -δε, y) + O(ε 2 ), ∂ n u ext (x 0 , y) = ∂ n u ext (x 0 + δε, y) -δε∂ 2 n u ext (x 0 + δε, y) + O(ε 2 ), ∂ n u int (x 0 , y) = ∂ n u int (x 0 -δε, y) + δε∂ 2 n u int (x 0 -δε, y) + O(ε 2 ).
Now we substitute these expressions in the boundary conditions over Γ of Equation ( 45) and neglecting the terms of order 2 or higher in ε, we obtain the new boundary conditions written over the new articial boundaries. The resulting asymptotic model writes as follows.

             σ int ∆u [1] δ,int = f int in Ω δ int , u [1] δ,int = ε(1 -2δ) 2 ∂ n u [1] δ,int on Γ δ int , u [1] δ,int = 0 on ∂Ω ∩ ∂Ω δ int .              σ ext ∆u [1] δ,ext = f ext in Ω δ ext , u [1] δ,ext = - ε(1 -2δ) 2 ∂ n u [1] δ,ext on Γ δ ext , u [1]
δ,ext = 0 on ∂Ω ∩ ∂Ω δ ext .

(

) 50
As we will see later in Section 4.3, with this new formulation, if we select δ > 1 2 , the negative term of the bilinear form in the variational formulation becomes positive and stability is restored. Henceforth, we will refer to this new stable model as stabilized δ-order 2 model. Notation 3. We denote by Ω δ the domain

Ω δ = Ω δ int ∪ Ω δ ext ,
where Ω δ int and Ω δ ext are the domains dened in Figure 3.

Classical conditions and comparison with equivalent conditions

In this section we show the results we obtain when we no longer consider the conductivity in the thin layer to be dependent on its thickness to remark the dierent results we obtain with our approach compared to this one. The model problem remains the same, we consider Equations (2) set in the domain showed in Figure 1, but know we consider a conductivity of the following form

σ =          σ int in Ω ε int , σ lay in Ω ε lay , σ ext in Ω ε ext ,
where σ lay is just a constant and not dependent on ε any more. With this conguration, the model problem we are considering is the one described by Equations [START_REF] Hughes | Isogeometric analysis: CAD, nite elements, NURBS, exact geometry and mesh renement[END_REF]. Considering this model, and applying the same asymptotic method developed in the previous section to derive approximate models, we obtain an rst-order model and a second-order model. The expression for these models are the following.

Inria

First-order model

                             σ int ∆u [0] int = f int in Ω int , σ ext ∆u [0] ext = f ext in Ω ext , u [0] Γ = 0, σ∂ n u [0] Γ = 0, u [0] = 0 on ∂Ω.
Second-order model

                               σ int ∆u [1] int = f int in Ω int , σ ext ∆u [1] ext = f ext in Ω ext , u [1] Γ = ε σ lay σ∂ n u [1] Γ -ε ∂ n u [1] Γ , σ∂ n u [1] Γ = -εσ lay d 2 dy 2 u [1] Γ -ε σ∂ 2 n u [1] Γ ,
u [1] = 0 on ∂Ω.

We notice that these models are dierent from ( 43) and [START_REF] Poignard | Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer[END_REF]. A main dierence comparing with our approach (i.e. when σ lay = σ 0 ε -3 ) comes from the fact that now both models are coupled, whereas employing our approach, the models are uncoupled, given thus by two independent problems. In this case, contrary to the rst class of ITCs, the order of these models, (3.4) and (3.4), coincide with the one of the models [START_REF] Perrussel | Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer[END_REF]

and (45). Stability and convergence results

This section is devoted to the validation of the multiscale expansions we have derived in the previous sections. We perform the proofs of existence, uniqueness and uniform estimates for the derived asymptotic models and we derive the convergence results. This section is structured as follows. In Section 4.1 we study the convergence of the asymptotic expansion for the reference model and give estimates for the residue. Then, Sections 4.2 and 4.3 are devoted to the study of the asymptotic models of the rst class and second class respectively. In these sections we derive the variational formulations for such asymptotic models and we prove stability and convergence results.

Convergence of the asymptotic expansion for the reference model

We begin by dening the residue for the reference Problem (2). We remind the domain Ω is dened in Section 1 and we can observe it in Figure 1. The residue of the asymptotic expansion ( 8) is dened by removing the rst terms to the solution u of the reference Problem (2). Denition 5. Let u be the solution to Problem (2), given the expansion in power series [START_REF] Caloz | Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism[END_REF] and a specic order N ∈ N, we dene the residue r N as

                             r N int (x, y) = u int (x, y) - N k=0 ε k u k int (x, y) in Ω ε int , r N ext (x, y) = u ext (x, y) - N k=0 ε k u k ext (x, y) in Ω ε ext , r N lay (x, y) = u lay (x, y) - N k=0 ε k U k x -x 0 ε , y in Ω ε lay .
Proposition 3. Let N ∈ N, the residue r N dened in Denition 5 satises the following equations

                                               σ int ∆r N int = 0 in Ω ε int , σ ext ∆r N ext = 0 in Ω ε ext , σ 0 ε -3 ∆r N lay = f N lay in Ω ε lay , r N int = r N lay on Γ ε int , r N lay = r N ext on Γ ε ext , σ 0 ε -3 ∂ n r N lay -σ int ∂ n r N int = g N int on Γ ε int , σ 0 ε -3 ∂ n r N lay -σ ext ∂ n r N ext = g N ext on Γ ε ext , r N = 0 on ∂Ω,
where Inria

f N lay = ε N -4 -σ 0 ∂ 2 y u N -1 lay -ε σ 0 ∂ 2 y u N lay , g N int = ε N -3 σ int ∂ n u N -3 int + . . . + ε 3 σ int ∂ n u N int , g N ext = ε N -4 σ ext ∂ n u N -3 ext + . . . + ε 3 σ ext ∂ n u N ext . (51)
Proof. We can deduce this result by applying Equations ( 9), ( 10), ( 11) and ( 3) to the denition of the residue.

Theorem 2. Let N ∈ N. For ε ∈ (0, ε 0 ) and under the assumptions

f N lay ∈ L 2 Ω ε lay , g N int ∈ L 2 (Γ ε int ), g N ext ∈ L 2 (Γ ε ext ), f N +5 lay ∈ L 2 Ω ε lay , g N +5 int ∈ L 2 (Γ ε int ), g N +5 ext ∈ L 2 (Γ ε
ext ), and u k ∈ H 1 (Ω ε ) for k ≤ N + 5, the following estimate holds for the residue dened in Denition 5,

r N ext 1,Ω ε ext + r N int 1,Ω ε int + √ ε r N lay 1,Ω ε lay ≤ Cε N +1 ,
for a positive constant C > 0 independent of ε.

Proof. Applying Theorem 1 and Proposition 3, for ε ∈ (0, ε 0 ), we deduce that there exists a unique r N ∈ H 1 0 (Ω) and r N 1,Ω = O(ε N -4 ). Rewriting r N in the following way

r N = r N +5 + ε N +1 u N +1 + ε N +2 u N +2 + ε N +3 u N +3 + ε N +4 u N +4 + ε N +5 u N +5 , we deduce that r N int 1,Ω ε int + r N ext 1,Ω ε ext = O(ε N +1 ). Finally taking into account that u k lay 1,Ω ε lay = O ε -1 2 , we deduce that r N lay 1,Ω ε lay = O ε N + 1 2
and the desired result.

Remark 3. We remark that the assumptions of Theorem 2 are rather strong assumptions because in general the expansion consumes regularity at each order and the rst term of the expansion u 0 , solution to [START_REF] Durué | Thin layer models for electromagnetism[END_REF], belongs only to H 3 (Ω ε ).

Validation of the rst class of ITCs

This section is devoted to the derivation of convergence results for the order 2 and order 4 asymptotic models we have derived in Section 2.2. We remark that the domain and conguration for these models have been presented in Section 1 and Figure 1.

Second order model: variational formulation

Problem ( 22) is uncoupled into two independent problems, therefore we write two variational formulations, one for each problem. First of all we introduce the functional spaces

H 1 0 (Ω ε int ) and H 1 0 (Ω ε ext ) as the functional framework. Assuming f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext ), the variational formulations reduce to nding u int ∈ H 1 0 (Ω ε int ) such that for all w int ∈ H 1 0 (Ω ε int ) - Ω ε int f int w int dx = Ω ε int σ int ∇u int • ∇w int dx, (52)
and nding

u ext ∈ H 1 0 (Ω ε ext ) such that for alld w ext ∈ H 1 0 (Ω ε ext ) - Ω ε ext f ext w ext dx = Ω ε ext σ ext ∇u ext • ∇w ext dx. (53)

Fourth order model: variational formulation

Instead of considering Problem ( 23) directly, we will consider the following problem

                           σ int ∆u int = f int in Ω ε int , σ ext ∆u ext = f ext in Ω ε ext , [u] Γ ε = 0, ε -2 σ 0 d 2 dy 2 {u} Γ ε + [σ∂ n u] Γ ε = g, u = 0 on ∂Ω ∩ ∂Ω ε . ( 54
)
This problem is similar to Problem ( 23) and it generalizes it by including the right-hand side function g. We begin by selecting the functional space denoted by V 4 and dened as follows Denition 6. The functional space V 4 is given by

V 4 = w : w int ∈ H 1 (Ω ε int ) , w ext ∈ H 1 (Ω ε ext ) , ∇ Γ ε {w} ∈ L 2 (Γ ε ) , w| Γ ε int = w| Γ ε ext , w| ∂Ω∩∂Ω ε int = 0, w| ∂Ω∩∂Ω ε ext = 0 As H 1 (Ω ε
) is a Hilbert space and V 4 is a closed subspace of H 1 (Ω ε ), we can deduce that the functional space V 4 , characterized in Denition 6 equipped with the norm

w V4 = w 2 1,Ω ε + ∇ Γ ε {w} 2 0,Γ ε 1 2 ,
is a Hilbert space. Remark 4. Mean values and jumps are dened over the interfaces Γ ε int and Γ ε ext . As jump and mean values only depend on the variable y, when we write Γ ε we are referring to the interval y ∈ (0, y 0 ).

Assuming f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext ), the variational formulation for problem 6 reduces to nding u ∈ V 4 , such that for all w ∈ V 4 , a(u, w) = l(w), (55)
where

a(u, w) = σ int Ω ε int ∇u • ∇w dx + σ ext Ω ε ext ∇u • ∇w dx + σ 0 ε -2 Γ ε ∇ Γ ε {u} Γ ε ∇ Γ ε {w} Γ ε ds, l(w) = - Ω ε int f int w dx - Ω ε ext f ext w dx - Γ ε g {w} Γ ε ds. Inria 4.2.

Stability results

First of all we will develop an expansion in power series of ε for the Problem [START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF] in the form

           u [3] ext ≈ k≥0 ε k u k ext in Ω ε int , u [3] int ≈ k≥0 ε k u k int in Ω ε ext . ( 56
)
We substitute these series in Equations ( 23) and we collect the terms with the same power in ε. For every k ∈ N we obtain the following set of equations

                           σ int ∆ u k int = f int δ 0 k in Ω ε int , σ ext ∆ u k ext = f ext δ 0 k in Ω ε ext , u k Γ ε = 0, -σ 0 d 2 dy 2 u k Γ ε = σ∂ n u k-2 Γ ε , u k = 0 on ∂Ω ∩ ∂Ω ε . ( 57
)
Denition 7. Given the expansion in power series [START_REF] Vial | Analyse multi-échelle et conditions aux limites approchées pour un problème avec couche mince dans un domaine à coin[END_REF] and N ∈ N, we dene the residue r N as

               r N int (x, y) = u [3] int (x, y) - N k=0 ε k u k int (x, y) in Ω ε int , r N ext (x, y) = u [3] ext (x, y) - N k=0 ε k u k ext (x, y)
in Ω ε ext .

We now prove the existence, uniqueness and uniform estimates of the solution to Problem (55), but before starting with the proofs of existence, uniqueness and estimates, we will write a Poincaré inequality for the conguration we are working on. Theorem 3. We can write a Poincaré inequality of the following form: there exists a constant C > 0 such that for all u ∈ V 4 ,

Ω ε |u| 2 dx ≤ C Ω ε |∇u| 2 dx.
Proof. We follow a similar reasoning to the one presented in [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF] (proposition 8.13 and corollary 9.19). For all u ∈ V 4 , u is continuous across the interfaces Γ ε int and Γ ε ext , and it vanishes in the rest of the boundary. We can deduce that for all (x, y)

∈ Ω ε |u(x, y)| ≤ I |∂ t u(t, y)| dt,
where we have dened I = 0, x 0 -ε 2 ∪ x 0 + ε 2 , L . We apply Cauchy-Schwartz inequality and by integrating rst in the x variable and then in the y variable, we obtain the desired result

Ω ε |u| 2 dx ≤ L 2 Ω ε |∂ x u| 2 dx ≤ L 2 Ω ε |∇u| 2 dx.
Theorem 4. For all ε > 0 there exists a unique u ∈ V 4 solution to Problem [START_REF] Solin | Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell's equations[END_REF] with data

f int ∈ L 2 (Ω ε int ) , f ext ∈ L 2 (Ω ε ext ) and g ∈ L 2 (Γ ε ).
Moreover, there exists ε 0 > 0 and a constant C > 0, such that for all ε ∈ (0, ε 0 ),

u V4 ≤ C f int 0,Ω ε int + f ext 0,Ω ε ext + g 0,Γ ε .
Proof. First of all, as the bilinear form a is coercive and continuous in V 4 , and the linear form l is continuous in V 4 , the existence and uniqueness of a weak solution follows from the Lax-Milgram Lemma. Then, for proving the uniform estimates, we select ε 0 as

ε 0 = σ 0 min(σ int , σ ext )
.

Then for ε < ε 0 and employing theorem 3, there exists a constant k 1 > 0 such that for all

w ∈ V 4 a(w, w) ≥ 1 k 1 min (σ int , σ ext ) w 2 V4 . (58)
Applying a trace theorem and Cauchy-Schwarz inequality to the denition of l we obtain that l is continuous in V 4 , more specically, there exists a constant k 2 > 0, such that

|l(w)| ≤ k 2 w V4 f int 0,Ω ε int + f ext 0,Ω ε ext + g 0,Γ ε . ( 59
)
Finally, employing Equations ( 58) and ( 59) we obtain

u V4 ≤ C f int 0,Ω ε int + f ext 0,Ω ε ext + g 0,Γ ε , where C = k 2 k 1 min (σ int , σ ext ) .
Proposition 4. Let N ∈ N, the residue r N dened in Denition 7 satises the following equations

                           σ int ∆ r N int = 0 in Ω ε int , σ ext ∆ r N ext = 0 in Ω ε
Proof. We can deduce this result by applying Equations ( 57) and ( 23) to the denition of the residue.

Theorem 5. Let N ∈ N. For ε ∈ (0, ε 0 ) and under the assumption g N ∈ L 2 (Γ ε ), the following estimate holds for the residue r N dened in Denition 7, for a constant

C > 0 independent of ε r N ext 1,Ω ε ext + r N int 1,Ω ε int ≤ Cε N +1 .
Proof. We can deduce this result directly from Theorem 4 and Proposition 4.

Convergence results

Theorem 6. Under the assumptions of Theorem 2 for N = 1, ε ∈ (0, ε 0 ) and with a data

f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext
), the following estimate holds for the function u [1] , solution to the second order asymptotic model [START_REF] Haddar | Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers[END_REF], which writes as

   σ int ∆u [1] int = f int in Ω ε int , u [1] int = 0 on ∂Ω ε int .    σ ext ∆u [1] ext = f ext in Ω ε ext , u [1] ext = 0 on ∂Ω ε ext .
and u, solution to the reference Problem (2): there exists a constant C > 0 independent of ε, such that

u int -u [1] int 1,Ω ε int + u ext -u [1] ext 1,Ω ε ext ≤ Cε 2 .
Proof. We can deduce this result directly from Theorem 2. Theorem 7. Under the assumptions of Theorem 2 and Theorem 5 for N = 3, ε ∈ (0, ε 0 ) and with a data f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext ), the following estimate holds for the function u [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF] , solution to the fourth order asymptotic model [START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF], which writes as

                               σ int ∆u [3] int = f int in Ω ε int , σ ext ∆u [3] ext = f ext in Ω ε ext , u [3] Γ ε = 0, σ∂ n u [3] Γ ε = - σ 0 ε 2 d 2 dy 2 u [3] Γ ε , u [3] = 0 on ∂Ω ∩ ∂Ω ε .
and u, solution to the reference Problem (2): there exists a constant C > 0 independent of ε, such that

u int -u [3] int 1,Ω ε int + u ext -u [3] ext 1,Ω ε ext ≤ Cε 4 .
Proof. To begin with, we consider the expansion (56) of u [3] we have done above. More specically, we consider Equations [START_REF] Vial | Eciency of approximate boundary conditions for corner domains coated with thin layers[END_REF]. We deduce that for k = 0, 1, 2, 3, we obtain

u k ≡ u k k = 0, 1 , 2, 3.
Thus, using Theorem 2 and Theorem 5 we deduce the desired result.

Validation of the second class of ITCs

This section is devoted to the derivation of convergence results for the rst-order and second-order asymptotic models we have derived in Section 3.2. We remind that the domain and conguration for these models have been presented in Section 3.1 and Figure 2.

First order model: variational formulation

Problem ( 43) is uncoupled into two independent problems, therefore we write two variational formulations, one for each problem. First of all we introduce the functional spaces H 1 0 (Ω int ) and H 1 0 (Ω ext ) as the functional framework. Assuming

f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the variational formulations reduce to nding u int ∈ H 1 0 (Ω int ) such that for all w int ∈ H 1 0 (Ω int ) - Ωint f int w int dx = Ωint σ int ∇u int • ∇w int dx, (60)
and nding

u ext ∈ H 1 0 (Ω ext ) such that for all w ext ∈ H 1 0 (Ω ext ) - Ωext f ext w ext dx = Ωext σ ext ∇u ext • ∇w ext dx. (61)

Second order model: variational formulation

In this section we derive a variational formulation for the second order asymptotic model [START_REF] Poignard | Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer[END_REF] we have derived in Section 3.2. We begin by selecting the functional spaces V int and V ext as the functional framework, which are dened as follows

V int = w ∈ H 1 (Ω int ) : w| ∂Ω∩∂Ωint = 0 , V ext = w ∈ H 1 (Ω ext ) : w| ∂Ω∩∂Ωext = 0 . (62)
Assuming

f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the variational formulations consist in nding u int ∈ V int , such that for all w int ∈ V int - Ωint f int w int dx = Ωint σ int ∇u int • ∇w int dx - Γ 2σ int ε u int w int ds,
and nding u ext ∈ V ext , such that for all

w ext ∈ V ext - Ωext f ext w ext dx = Ωext σ ext ∇u ext • ∇w ext dx - Γ 2σ ext ε u ext w ext ds.
Observing these variational formulations we notice that we cannot prove the coerciveness of the bilinear forms due to the terms

- Γ 2σ int ε u int w int ds and - Γ 2σ ext ε u ext w ext ds
Inria being negative. These negative terms could cause instabilities when numerically solving the problem with the nite element method. However, to overcome this problem and recover stability, we have derived new models across some articial boundaries in Section 3.3. With these models we will no longer have instability problems, as we will prove in the following section.

Stabilized δ-order 2 model: variational formulation

In this section we derive a variational formulation for the stabilized δ-order 2 asymptotic model [START_REF] Schmidt | A unied analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model[END_REF] we have derived in Section 3.3. Instead of directly considering Problem (50), we will consider the following problem

             σ int ∆u int = f int in Ω δ int , u int - ε(1 -2δ) 2 ∂ n u int = g int on Γ δ int , u int = 0 on ∂Ω ∩ ∂Ω δ int .              σ ext ∆u ext = f ext in Ω δ ext , u ext + ε(1 -2δ) 2 ∂ n u ext = g ext on Γ δ ext , u ext = 0 on ∂Ω ∩ ∂Ω δ ext .
This problem is similar to Problem ( 50) and it generalizes it by including the right-hand side functions g 1 and g 2 . We begin by selecting the functional framework. We introduce the functional spaces V δ int and V δ ext , which are dened as follows

V δ int = w ∈ H 1 Ω δ int : w| ∂Ω∩∂Ω δ int = 0 , V δ ext = w ∈ H 1 Ω δ ext : w| ∂Ω∩∂Ω δ ext = 0 . Assuming f int ∈ L 2 Ω δ int and f ext ∈ L 2 Ω δ ext , the variational formulations reduce to nding u int ∈ V δ int , such that for all w int ∈ V δ int a int (u int , w int ) = l int (w int ), (63)
and nding

u ext ∈ V δ ext , such that for all w ext ∈ V δ ext a ext (u ext , w ext ) = l ext (w ext ), (64)
where

a int (u int , w int ) = Ω δ int σ int ∇u int • ∇w int dx - Γ δ int 2σ int ε(1 -2δ) u int w int d2, a ext (u ext , w ext ) = Ω δ ext σ ext ∇u ext • ∇w ext dx - Γ δ ext 2σ ext ε(1 -2δ) u ext w ext ds, and
l int (w int ) = - Ω δ int f int w int dx - Γ δ int 2σ int ε(1 -2δ) g int w int ds, l ext (w ext ) = - Ω δ ext f ext w ext dx - Γ δ ext 2σ ext ε(1 -2δ) g ext w ext ds.
Now, with these variational formulations, we can observe that if we select δ > 1 2 , the last terms of the bilinear forms will be positive and thus enforce the coerciveness of the corresponding bilinear forms.

Stability results

First of all we will develop an expansion in power series of ε for the Problem (50) in the form

           u [1] δ,ext ≈ k≥0 ε k u k δ,ext in Ω δ int , u [1] δ,int ≈ k≥0 ε k u k δ,int in Ω δ ext . (65)
We substitute these series into the Equations ( 50) and we collect the terms with the same powers in ε. For every k ∈ N we obtain the following set of equations

         σ int ∆ u k δ,int = f int δ 0 k in Ω δ int , u k δ,int = ε(1 -2δ) 2 ∂ n u k-1 δ,int in Γ δ int , u k δ,int = 0 on ∂Ω ∩ ∂Ω δ int .          σ ext ∆ u k δ,ext = f ext δ 0 k in Ω δ ext , u k δ,ext = - ε(1 -2δ) 2 ∂ n u k-1 δ,ext in Γ δ ext , u k δ,ext = 0 on ∂Ω ∩ ∂Ω δ ext . (66)
Denition 8. Given the expansion in power series (65) and N ∈ N, we dene the residue r N δ as

               r N δ,int (x, y) = u [3] δ,int (x, y) - N k=0 ε k u k δ,int (x, y), r N δ,ext (x, y) = u [3] δ,ext (x, y) - N k=0 ε k u k δ,ext (x, y).
We now prove the existence, uniqueness and uniform estimates of the solution to Problems (63) and (64). Theorem 8. For all ε > 0 and δ > 1 2 there exists a unique u = (u int , u ext ) where u int ∈ V δ int and

u ext ∈ V δ
ext are solutions to (63) and (64) respectively, with data

f int ∈ L 2 (Ω δ int ), f ext ∈ L 2 (Ω δ ext ), g int ∈ L 2 (Γ δ int ), g ext ∈ L 2 (Γ δ ext ).
Moreover there exists ε 0 and a constant C > 0 such that for all

ε ∈ (0, ε 0 ) Inria u 1,Ω δ ≤ ε -1 C f int 0,Ω δ int + f ext 0,Ω δ ext + g int 0,Γ δ int + g ext 0,Γ δ ext . (67)
Proof. First of all as the bilinear form a int is coercive and continuous in V δ int and the linear form l int is also continuous in V δ int , the existence and uniqueness of a weak solution follows from the Lax-Milgram Lemma. Then, for proving the uniform estimates, employing Poincaré inequality, there exists a constant k 1 such that for all

w ∈ V δ int a int (w, w) ≥ σ int k 1 w 2 1,Ω δ int . ( 68
)
Applying a trace theorem for the Dirichlet trace operator γ Γ δ int we have that there exists a constant k 2 such that

γ Γ δ int (w) 1 2 ,Γ δ int ≤ k 2 w 1,Ω δ int .
We select ε 0 as

ε 0 = 2k 2 σ int 2δ -1 .
Then, for ε ∈ (0, ε 0 ), and applying Cauchy-Schwarz inequality to the denition of l int we obtain that there exists a constant k 3 , such that

|l int (w)| ≤ ε -1 k 3 w 1,Ω δ int f int 0,Ω δ int + g int 0,Γ δ int . (69)
Finally, employing Equations ( 68) and (69) we obtain

u int 1,Ω δ int ≤ ε -1 C f int 0,Ω δ int + g int 0,Γ δ int , where C = k 1 k 2 σ int .
The same proof holds for the equations in Ω δ ext , and employing these two results we obtain the desired result. Proposition 5. Let N ∈ N, the residue r N δ dened in Denition 8 satises the following equations

             σ int ∆ r N δ,int = 0 in Ω δ int , r N δ,int - ε(1 -2δ) 2 r N δ,int = g N int on Γ δ int , r N δ,int = 0 on ∂Ω ∩ ∂Ω δ int ,              σ int ∆ r N δ,ext = 0 in Ω δ ext , r N δ,ext + ε(1 -2δ) 2 r N δ,ext = g N ext on Γ δ ext , r N δ,ext = 0 on ∂Ω ∩ ∂Ω δ ext ,
where

g N int = ε N +1 1 -2δ 2 ∂ n u N int , g N ext = ε N +1 1 -2δ 2 ∂ n u N ext .
Proof. We can deduce this result by applying Equations ( 66) and ( 50) to the denition of the residue.

Theorem 9. Let N ∈ N. For ε ∈ (0, ε 0 ) and under the assumptions

g N int ∈ L 2 (Γ δ int ), g N ext ∈ L 2 (Γ δ ext ), g N +1 int ∈ L 2 (Γ δ int ), g N +1 ext ∈ L 2 (Γ δ
ext ), functions dened in Proposition 5, and u k ∈ H 1 (Ω ε ) for k ≤ N + 1, there exists a constant C > 0, independent of ε, for which the following estimate holds for the residue r N δ dened in Denition 8,

r N δ,ext 1,Ω δ ext + r N δ,int 1,Ω δ int ≤ Cε N +1 .
Proof. From Theorem 8 and Proposition 5 we deduce that

r N δ,ext 1,Ω δ ext + r N δ,int 1,Ω δ int = O(ε N ).
Finally, writing

   r N δ,int = r N +1 δ,int + ε N +1 u N +1 δ,int , r N δ,ext = r N +1 δ,ext + ε N +1 u N +1
δ,ext , we deduce the desired result.

Convergence results

Theorem 10. Under the assumptions of Theorem 2 for N = 0 and ε ∈ (0, ε 0 ) , with the data f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the following estimate holds for the function u [0] solution to the rst order asymptotic model [START_REF] Perrussel | Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer[END_REF], which writes as

   σ int ∆u [0] int = f int in Ω int , u [0] int = 0 on ∂Ω int ,    σ ext ∆u [0] ext = f ext in Ω ext , u [0] ext = 0 on ∂Ω ext .
and u, solution to the reference Problem (2): there exists a constant C > 0 independent of ε, such that

u int -u [0] int 1,Ω ε int + u ext -u [0] ext 1,Ω ε ext ≤ Cε.
Proof. We can deduce this result directly from Theorem 2. Inria Theorem 11. Under the assumptions of Theorem 2 and Theorem 9 for N = 1 and ε ∈ (0, ε 0 ), with the data f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the following estimate holds for u [1] δ , solution to the stabilized δ-order 2 asymptotic model [START_REF] Schmidt | A unied analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model[END_REF], which writes as

             σ int ∆u [1] δ,int = f int in Ω δ int , u [1] δ,int = ε(1 -2δ) 2 ∂ n u [1] δ,int on Γ δ int , u [1]
δ,int = 0

on ∂Ω ∩ ∂Ω δ int .              σ ext ∆u [1] δ,ext = f ext in Ω δ ext , u [1] δ,ext = - ε(1 -2δ) 2 ∂ n u [1] δ,ext on Γ δ ext , u [1]
δ,ext = 0

on ∂Ω ∩ ∂Ω δ ext .
and u, solution to the reference Problem (2): there exists a constant C > 0 independent form ε, such that

u int -u [1] δ,int 1,Ω δ int + u ext -u [1] δ,ext 1,Ω δ ext ≤ Cε 2 .
Proof. To begin with, we consider the expansion (65) of u [1] δ we have obtained above. More specically, we consider Equations (66). We truncate the series from the second term and we consider the truncated series u

(1)

δ = u 0 δ + ε u 1 δ .
We rewrite the equations of Problem [START_REF] Poignard | Rigorous asymptotics for the elecric eld in TM mode at mid-frequency in a bidimensional medium with thin layer[END_REF] to derive the conditions on the articial boundaries Γ δ int and Γ δ ext for u

δ following the same procedure as in Section 3.3. If we apply Theorem 8, we deduce

u (1) δ -u (1) δ 1,Ω δ ≤ Cε 2 .
Finally applying Theorem 2 and Theorem 9 we deduce the desired result. Numerical results

The previous sections were devoted to the derivation and analysis of several asymptotic models. In this section we numerically asses the performance of these models and we check if the obtained numerical results match the theoretical results. For obtaining such numerical results, a Finite Element Method has been implemented employing Matlab and C programming language. The code could be divided into two main parts. The rst one consists in the assembling of a linear system and the second one consists in solving such linear system. The part corresponding to the assembling is mainly coded in C and the part corresponding to the resolution of the linear system is mainly coded in Matlab. Moreover, Matlab is employed for the post processing of the solution, including tasks like the visualization of the solution or the calculus of the error in dierent norms.

This implementation corresponds to the classical Finite Element Method. It is based on straight triangular elements for discretizing the domain of the problem and piecewise polynomials of any given degree for representing the solution. Such polynomials correspond to the Lagrange interpolating Polynomials. We mainly employ structured meshes like the one showed in Figure 4 due to the considered domains being mainly rectangles, but the code is adapted to work with unstructured meshes generated with other mesh generators. The problem we are interested in solving is the Poisson's equation σ∆u = f, where the conductivity σ and the right-hand side function f are known data. In general this equation is set in a domain Ω ⊂ R 2 , composed of several subdomains and the conductivity is considered to be a piecewise constant function which takes a dierent value inside each subdomain. The code is capable of dealing with this kind of complex conguration, as well as with the dierent transmission conditions required to solve some of the derived asymptotic models.

First class of ITCs

In this section we show some numerical tests regarding the approximate problems we have derived. For this purpose, we use the Finite Element Method along with the variational formulations derived in Sections 1, 4.2 and 4.3 to obtain approximate solutions of these models. The objective of these numerical tests is to illustrate the theory by checking if the theoretical orders of convergence coincide with the ones obtained numerically. For these experiments, the Inria considered domain is a 2 m × 1 m rectangle domain. In addition, we consider a conductivity of the following form

σ =          σ int = 5 S/m in Ω ε int , σ lay = 8.89ε -3 S/m in Ω ε lay , σ ext = 3 S/m in Ω ε
ext , where ε represents the thickness of the thin layer, and the following right-hand side

f =          f int = 1 C in Ω ε int , f lay = 0 C in Ω ε lay , f ext = 1 C in Ω ε ext .
where the units S/m and C correspond to Siemens per meter and Coulomb respectively. With these congurations and parameters, we solve the reference Problem (2) and asymptotic models ( 22) and ( 23) by employing the Finite Element Method. For discretizing the domain, we have used 384 triangular shaped elements and Lagrange interpolating Polynomials of second degree.

We begin by performing a qualitative comparison between the solution to the reference model ( 2) and the approximate models [START_REF] Haddar | Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers[END_REF] and [START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF]. Employing the stated parameters and conguration, we have solved these models and their solutions are shown in Figure 5. We notice that the fourth-order model is more accurate and approximates better the eect of the high conductive thin layer than the second-order model.

We calculate the H 1 errors between the solution to the reference model ( 2) and the approximate models ( 22) and ( 23) for dierent thicknesses of the thin layer. These results can be observed in Figure 6, where we show the obtained convergence rates for the H 1 relative error. From these results we observe that the numerical convergence rates we obtain coincide with the theoretical convergence rates proved in Section 4.2.

Second class of ITCs

In the same way we have done in the previous section, we would like to begin by performing a qualitative comparison of the solution to the reference model and the asymptotic models of the second class. We have employed the same physical parameters as in the previous section for these new tests. In Figure 7, we observe the solution we obtain for the reference Problem (2) and the asymptotic models of order 1 (43) and order 2 (45) of the second class.

As we stated in the previous sections, the stability of the second-order model cannot be guaranteed, this fact induces big changes in the solution when we slightly change the parameters of the problem. We illustrate this fact in Figure 8, where we show the solution for the secondorder model for dierent values of ε. We observe that the solution drastically changes around the transmission conditions for every small change in the value of ε.

To deliver a more quantitative comparison of the models we have derived. We have calculated the H 1 error between the reference solution (2) and the approximate models [START_REF] Perrussel | Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer[END_REF] and [START_REF] Poignard | Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer[END_REF] for dierent thicknesses of the thin layer. Before showing the H 1 norm results, we show similar results regarding the L 2 norm to remark how the second-order model behaves dierently depending on the norm we choose. In Figure 9 we observe the obtained convergence rates for the L 2 absolute and relative errors and in Figure 10 the convergence rates for the H 1 absolute and relative errors.

We observe that the numerical convergence rates we have obtained coincide with the theoretical convergence rates proved in Section 4.3 for the rst-order model. On the other hand, for the second-order model, even though we still recover the expected theoretical order of convergence for the L 2 absolute and relative errors, due to the instabilities it does not perform as well for the H 1 absolute and relative errors. Even so, it still outperforms the rst-order model.

To face these instabilities, in Section 3.3, we have derived a new second-order model by dening some articial conditions and moving the boundary conditions to these new boundaries.

To derive this new model [START_REF] Schmidt | A unied analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model[END_REF] we employ a parameter δ that controls the distance between the articial boundaries. In Section 4.3.3 we prove that for δ > 0.5 this approach solves the problem of instabilities. Figure 11 shows a problem with instabilities and how they can be eliminated when applying a δ parameter which is greater than 0.5. However, the instabilities are not completely removed if it is not greater than 0.5. To illustrate this fact, the example of Figure 11 shows that for δ=0.1 the instabilities are still present, whereas when δ = 0.51 is applied, we do not have instabilities any more.

In Figure 9, we compare the obtained convergence rates for the L 2 relative error and in Figure 10 the convergence rates for the H 1 relative error for the unstable order 2 model and for the stabilized δ-order 2 model. We observe that for the L 2 error both models behave similarly but for the H 1 error, the second-order model does not converge properly, whereas the stabilized δ-order 2 model delivers the correct convergence rates.

From these results we observe that if we apply the articial boundary approach with a δ greater than 0.5, we have no longer instability problems and the numerical convergence rates coincide with the theoretical convergence rates proved in Section 4.3, for both the L 2 and the H 1 errors.

Comparison between the rst and the second class

In this section we will do a brief comparison between the asymptotic models we have derived, mentioning the strong and weak points of each class. Regarding the convergence, considering the rst class of ITCs, the model with highest order reaches a convergence of order 4, whereas for the second class, the model with highest order only reaches a convergence of order 2. We can observe these convergence rates in L 2 norm for the ve models we have derived in Figure 12 and in H 1 norm in Figure 13. We see that all the models converge with the expected order of Another drawback of the second class of ITCs is that the model of order 2 presents instabilities, whereas the models derived for the rst class are both stable.

Regarding the domain, a strong point of the second class is that the domain does not depend on ε, while the domain for the rst class depends on ε. Even though this point is not very Inria relevant for our conguration, due to the thin layer having a straight shape, it could be very interesting when considering more complex congurations, in which the shape of the thin layer is curved. In such a case, the fact of having a single interface between the two subdomains instead of having a gap greatly reduces the numerical complexity of the model. All these features are summarized in table 1

   (σ int -i 0 ω) ∆u [1] int = f int in Ω ε int , u [1] int = 0 on ∂Ω ε int .    (σ ext -i 0 ω) ∆u [1] ext = f ext in Ω ε ext , u [1] ext = 0 on ∂Ω ε ext . (72)
Assuming

f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext ), the variational formulations consist in nding u int ∈ H 1 0 (Ω ε int ), such that for all w int ∈ H 1 0 (Ω ε int ) - Ω ε int f int w int dx = Ω ε int (σ int -i 0 ω) ∇u int • ∇w int dx,
and nding

u ext ∈ H 1 0 (Ω ε ext ) such that for all w ext ∈ H 1 0 (Ω ε ext ) - Ω ε ext f ext w ext dx = Ω ε ext (σ ext -i 0 ω) ∇u ext • ∇w ext dx. A.1.3 First class: fourth-order model                                (σ int -i 0 ω) ∆u [3] int = f int in Ω ε int , (σ ext -i 0 ω) ∆u [3] ext = f ext in Ω ε ext , u [3] Γ ε = 0, (σ -i 0 ω) ∂ n u [3] Γ ε = - σ 0 ε 2 d 2 dy 2 u [3] Γ ε , u [3] = 0 on ∂Ω ∩ ∂Ω ε . (73)
Assuming

f int ∈ L 2 (Ω ε int ) and f ext ∈ L 2 (Ω ε ext ), the variational problem reduces to nding u ∈ V 4 , such that for all w ∈ V 4 , (σ int -i 0 ω) Ω ε int ∇u • ∇w dx + (σ int -i 0 ω) Ω ε ext ∇u • ∇w dx + σ 0 ε -2 Γ ε ∇ Γ ε {u} Γ ε ∇ Γ ε {w} Γ ε ds = - Ω ε int f int w dx - Ω ε ext f ext w dx.
A.1.4 Second class: rst-order model

   (σ int -i 0 ω) ∆u [0] int = f int in Ω int , u [0] int = 0 on ∂Ω int .    (σ ext -i 0 ω) ∆u [0] ext = f ext in Ω ext , u [0] ext = 0 on ∂Ω ext . (74)

RR n°8998

Assuming f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the variational formulations of these problems consist in nding u int ∈ H 1 0 (Ω int ) such that for all w int ∈ H 1 0 (Ω int )

- Ωint f int w int dx = Ωint (σ int -i 0 ω) ∇u int • ∇w int dx,
and nding

u ext ∈ H 1 0 (Ω ext ) such that for all w ext ∈ H 1 0 (Ω ext ) - Ωext f ext w ext dx = Ωext (σ ext -i 0 ω) ∇u ext • ∇w ext dx.
A.1.5 Second class: second-order model

             (σ int -i 0 ω) ∆u [1] int = f int in Ω int , u [1] int = ε 2 ∂ n u [1] int on Γ, u [1] int = 0 on ∂Ω ∩ ∂Ω int .              (σ ext -i 0 ω) ∆u [1] ext = f ext in Ω ext , u [1] ext = - ε 2 ∂ n u [1] ext on Γ, u [1]
ext = 0 on ∂Ω ∩ ∂Ω ext .

(75)

Assuming f int ∈ L 2 (Ω int ) and f ext ∈ L 2 (Ω ext ), the variational formulations reduce to looking for u int ∈ V int , such that for all w int ∈ V int where the spaces V int and V ext have been dened in (62).

Remark 5. The models obtained in this section are very similar to the ones obtained for the static case, [START_REF] Haddar | Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers[END_REF] and [START_REF] Hernandes | Electric potential due to an innite conducting cylinder with internal or external point charge[END_REF] for the rst class and [START_REF] Perrussel | Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer[END_REF] and [START_REF] Poignard | Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer[END_REF] for the second class. In fact, it is possible to obtain the models we present in this section by simply substituting the conductivities σ int and σ ext by σ int -i 0 ω and σ ext -i 0 ω respectively.

A.2 3D axisymmetric conguration

The main objective of this section is the derivation of approximate models in a 3D axisymmetric conguration. The plan of the section is the following. First we set the model problem we are interested in. Then, we develop a multiscale expansion in powers of ε for the solution to the model problem and we obtain the equations for the rst terms of the expansion adopting the rst approach. Finally, we derive the desired approximate models. We then address the second class of problems, and for avoiding repetition with the previous sections only the main results are presented.

Inria

A.2.1 Model problem and scaling

Let Ω ⊂ R 3 be the domain of interest described at Figure 14. The Domain Ω is a cylinder shaped domain and is decomposed into three subdomains: Ω ε int , Ω ε ext , and Ω ε lay . Subdomain Ω ε lay is a thin layer of uniform thickness ε > 0. We denote by Γ ε int the interface between Ω ε int and Ω ε lay , and by Γ ε ext the interface between Ω ε lay and Ω ε ext . In this domain, we study the static electric potential equation, which read as follows div(σ∇u) = f.

(76) z y

x Here, u represents the electric potential, σ stands for the conductivity and f is the right-hand side, which corresponds to a current source. The conductivity is a piecewise constant function, with a dierent value in each subdomain. Specically, the value of the conductivity inside the thin layer Ω ε lay is much larger than the one in the other subdomains and we assume that it depends on parameter ε. We consider a conductivity of the following form

Ω ε lay Ω ε int Ω ε ext Γ ε ext Γ ε int Γ n n ε
σ =          σ int in Ω ε int , σ lay = σ 0 ε -3 in Ω ε lay , σ ext in Ω ε ext ,                 
u k (R 0 , θ, z) = 0 (θ, z) ∈ [0, 2π) × (0, z 0 ), u k (r, θ, 0) = u k (r, θ, z 0 ) = 0 (r, θ) ∈ 0, r 0 -

ε 2 ∪ r 0 + ε 2 , R 0 × [0, 2π), U k (R, θ, 0) = U k (R, θ, z 0 ) = 0 (R, θ) ∈ - 1 2 , 1 2 × [0, 2π). (85a) (85b) (85c)
For determining the elementary problem satised by each of the terms of the expansion, we will also need the following equation obtained by applying the fundamental theorem of calculus for a smooth function U k ,

1 2 -1 2 ∂ 2 R U k (R, z) dR = ∂ R U k 1 2 , z -∂ R U k - 1 2 , z .
If we substitute Equation (83) to the left-hand side and Equations (84c) and (84d) to the righthand side, we obtain the following compatibility condition

1 2 -1 2 ∂ 2 z U k-2 (R, θ, z) + k-1 l=0 (-R) k-1-l r k-l 0 ∂ R U l (R, θ, z) + k-2 l=0 (k -l -1) (-R) k-2-l r k-l 0 ∂ 2 θ U l (R, θ, z) dR = -1 σ 0 σ∂ n u k-4 Γ ε (z). (86)
We adopt the convention that the terms with negative indices in Equations ( 82)-( 86) are equal to 0. Employing Equations ( 82) -(86) we deduce the elementary problems satised outside and inside the layer for any k ∈ N. For that purpose we employ the following algorithm composed of three steps.

Algorithm for the determination of the coecients

Initialization of the algorithm:

Before showing the dierent steps to obtain function U k and u k for every k, we need to determine function U 0 up to a function in the variables θ and z, denoted by ϕ 0 0 . For that purpose we consider Equations (83), (84c), and (84d), and we build the following dierential problem in the variable R for U 0 (the variables θ and z play the role of a parameter)

                     ∂ 2 R U 0 (R, θ, z) = 0 R ∈ - 1 2 , 1 2 , σ 0 ∂ R U 0 - 1 2 , θ, z = 0, σ 0 ∂ R U 0 1 2 , θ, z = 0.
From these equations we deduce that U 0 has the following form U 0 (R, θ, z) = ϕ 0 0 (θ, z), where function ϕ 0 0 has yet to be determined and this will be done during the rst step of the algorithm. After these preliminary steps, we move onto determining U k and u k for any k.

We assume that the rst terms of the expansion (81) up to the order ε k-1 have already been calculated and we calculate the equations for the k-th term. We also assume that at rank k we know the form of U k up to a function in the variables θ and z, denoted by ϕ k 0 . We obtain the expression of U k at rank k -1. The rst step consists in determining the expression of function U k+1 up to function ϕ k+1 0 . Then, at the second step we determine function ϕ k 0 involved in the expression of function U k . Finally, we determine u k int and u k ext at the third step. For every k = 0, 1, 2, . . ., we perform the following steps: First step:

We select Equations ( 83), (84c), and (84d), and we build the following dierential problem in the variable R for U k+1 (the variables θ and z play the role of a parameter)

                     ∂ 2 R U k+1 (R, θ, z) = g k+1 (R, θ, z) R ∈ - 1 2 , 1 2 , σ 0 ∂ R U k+1 - 1 2 , θ, z = σ int ∂ n u k-3 int r 0 - ε 2 , θ, z , σ 0 ∂ R U k+1 1 2 , θ, z = σ ext ∂ n u k-3 ext r 0 + ε 2 , θ, z , (87)
where

g k+1 (R, θ, z) = - k l=0 (-R) k-l r k-l+1 0 ∂ R U l (R, θ, z)- k-1 l=0 (k-l) (-R) k-l-1 r k-l+1 0 ∂ 2 θ U l (R, θ, z)-∂ 2 z U k-1 (R, θ, z) .
There exists a solution U k+1 to (87) provided the compatibility condition (86) is satised. We deduce the expression of U k+1 up to a function in the variables θ and z, denoted by ϕ k+1 0 (θ, z). The function U k+1 has the following form U k+1 (R, θ, z) = V k+1 (R, θ, z) + ϕ k+1 0 (θ, z), where V k+1 represents the part of U k+1 that is determined at this step and has the form (see Proposition 6)

V k (R, θ, z) =    0 if k = 0, 1, 2, 3 ϕ k k-2 (θ, z)R k-2 + ϕ k k-3 (θ, z)R k-3 + . . . + ϕ k 1 (θ, z)R if k > 3.
Function ϕ k+1 0 represents the part of U k+1 that is determined at the following rank.

Second step:

We employ the compatibility condition (86) (at rank k + 2), along with Equation (85c) to write the following dierential problem in the variables θ and z for function ϕ k 0 , present in the expression of U k . We conclude that ϕ 0 0 (θ, z) = 0 and thus, U 0 (R, θ, z) = 0. Finally, employing (89), we obtain that the limit solution u 0 satises homogeneous Dirichlet boundary conditions on Γ ε int and Γ ε ext . Thus, the problem satised by u 0 reads as

            
   σ int ∆u 0 int = f int in Ω ε int , u 0 int = 0 on ∂Ω ε int .    σ ext ∆u 0 ext = f ext in Ω ε ext , u 0 ext = 0 on ∂Ω ε ext .
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Terms of order one

We consider Problem (87) for

U 2                      ∂ 2 R U 2 (R, θ, z) = 0 R ∈ -1 2 , 1 2 , ∂ R U 2 - 1 2 , θ, z = 0, ∂ R U 2 1 2
, θ, z = 0.

We deduce that the solution to this equation has the form U 2 (R, θ, z) = ϕ 2 0 (θ, z). Then, we employ (88) and we obtain the following problem for ϕ 1 0

             ∂ 2 z ϕ 1 0 (θ, z) + 1 r 2 0
∂ 2 θ ϕ 1 0 (θ, z) = 0 (θ, z) ∈ [0, 2π) × (0, z 0 ), ϕ 1 0 (0, θ) = 0, ϕ 1 0 (z 0 , θ) = 0.

We conclude that ϕ 1 0 (θ, z) = 0 and thus, U 1 (R, θ, z) = 0. Finally, employing (89) we write the problem satised outside the layer by u 1 as two uncoupled problems

Figure 1 :

 1 Figure 1: Domain of interest, composed of a thin layer, an interior domain and an exterior domain.

  Inria

3

 3

  a) Reference problem domain. Domain for the second class of ITCs.

Figure 2 :

 2 Figure 2: Domains for the reference model and the second class of asymptotic models.

Figure 3 :

 3 Figure 3: New conguration for the domain composed of two articial boundaries.

Figure 4 :

 4 Figure 4: Structured mesh for a rectangular domain.

  Reference model, x-y axis view. Reference model, x-z axis view. Second-order model, x-y view. Fourth-order model, x-y view.

  Order4 model, x-z view.

Figure 5 :

 5 Figure 5: Solution to the reference Problem (2), the second-order model (22) and the fourth-order model (23) of the rst class.Inria

Figure 6 :

 6 Figure 6: H 1 relative error of the second-order model (22) and fourth-order model (23) of the rst class for dierent values of ε.

  Second-order model, x-z view.

Figure 7 :

 7 Figure 7: Solution to the reference Problem (2), the rst-order model (43) and the second-order model (45) of the second class.Inria

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Instabilities of the solutions to the second-order model (45) for dierent values of ε.

Figure 11 :

 11 Figure 11: Removing the instabilities of the order 2 model (45) with the articial boundaries.

Figure 13 :

 13 Figure 13: H 1 relative error of the dierent asymptotic models for dierent values of ε

-f

  Ωint int w int dx = Ωint (σ int -i 0 ω) ∇u int • ∇w int dx -Γ 2 (σ int -i 0 ω) ε u int w int ds,and looking for u ext ∈ V ext , such that for allw ext ∈ V ext -Ωext f ext w ext dx = Ωext (σ ext -i 0 ω) ∇u ext • ∇w ext dx -Γ 2 (σ ext -i 0 ω) ε u ext w ext ds,

Figure 14 :

 14 Figure 14: Sectioned three dimensional domain for the model problem and asymptotic models of the rst class.

  z) = h k (θ, z) (θ, z) ∈ [0, 2π) × (0, z 0 ), ϕ k 0 (θ, 0) = 0 θ ∈ [0, 2π), ϕ k 0 (θ, z 0 ) = 0 θ ∈ [0, 2π),(88)Inria employ (88) and we build the following problem for ϕ z) ∈ [0, 2π) × (0, z 0 ),

Figure 15 :

 15 Figure 15: Sectioned domain for the asymptotic models of the second class.

  accuracy in L 2 norm. On the other hand, in H 1 norm, all models converge with the expected order of accuracy except the second-order model of the second class due to the instabilities. Figure 12: L 2 relative error of the dierent asymptotic models for dierent values of ε

			Class 2: Order 1	Class 2: Order 2	δ-Order 2
			Class 1: Order 2	Class 1: Order 4
		10 -2
	Relative Error	10 -6 10 -4
	L 2
		10 -8
		0.0039	0.0066	0.0110	0.0186	0.0312
			Casing Thickness (m)
			Class 2: Order 1	Class 2: Order 2	δ-Order 2
			Class 1: Order 2	Class 1: Order 4
		10 -2
	Relative Error	10 -6 10 -4
	H 1
		10 -8
		0.0039	0.0066	0.0110	0.0186	0.0312
			Casing Thickness (m)

Table 1 :

 1 . Comparison of the dierent derived models.

	Model	Numerical order Stability ε-independent domain
	Class 1: Order 2	2
	Class 1: Order 4	4
	Class 2: Order 1	1
	Class 2: Order 2	1-2
	δ-Order 2	2

RR n°8998

ext , r N Γ ε = 0, ε -2 σ 0 d 2 dy 2 r N Γ ε + σ∂ n r N Γ ε = g N , r N = 0on ∂Ω ∩ ∂Ω ε , whereg N = ε N +1 -σ∂ n u N +1 Γ ε -ε σ∂ n u N +2 Γ ε .Inria

avenue de la Vieille Tour 33405 Talence Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

H. Barucq, A. Erdozain & V.Péron Inria ITC's for the Electric Potential

Appendix A Additional results

A. [START_REF] Bendali | The eect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation[END_REF]

Fourth-order model

For deriving the fourth-order model, we truncate the series from the fourth term and we dene u (3) as u (3) = u 0 + εu 1 + ε 2 u 2 + ε 3 u 3 in Ω int ∪ Ω ext (see Proposition 2).

From [START_REF] Pérez-Arancibia | On the Green's function for the Helmholtz operator in an impedance circular cylindrical waveguide[END_REF], [START_REF] Péron | Mathematical modeling of electromagnetic phenomena in high contrast media[END_REF], [START_REF] Péron | Equivalent boundary conditions for an elasto-acoustic problem set in a domain with a thin layer[END_REF] and [START_REF] Péron | Equivalent transmission conditions for the timeharmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet[END_REF] we can deduce that u (3) satises the following equations

int (x 0 , y) = g 3 1 (y) y ∈ (0, y 0 ), u

int = 0 on ∂Ω ∩ ∂Ω int .

ext (x 0 , y) = g 3 2 (y) y ∈ (0, y 0 ), u

ext = 0 on ∂Ω ∩ ∂Ω ext .

(

where g 3 1 and g 3 2 are dened as follows Appendix A Additional results

In this section we present asymptotic models for similar congurations to the one presented in the previous sections. The congurations considered in this section include a 2D conguration for a time-harmonic problem, and a 3D axisymmetric borehole shaped conguration. Due to similarities in the procedure, in this section we will concentrate in presenting the resulting asymptotic models.

A.1 Time-harmonic problem

Model problem

Here we consider again the equation for the electric potential, but now we consider the frequency to be non-zero. The problem writes as follows

where u represents the electric potential, σ stands for the conductivity, f denotes a current source, ω is the frequency and 0 is the permittivity. We consider the same domain we had in Section 1, which is depicted at Figure 1. We consider the conductivity to be piecewise constant and to have a dierent value in each subdomain, being of the form σ lay = σ 0 ε -3 inside the thin layer. Both the right-hand side f and the conductivity σ have the same form as the ones considered in Section 1. In this framework, the Problem (70) writes as follows

where ∂ n represents the normal derivative in the direction of the normal vector, inwardly directed to Ω ε ext on Γ ε ext , and outwardly directed to Ω ε int on Γ ε int , see Figure 1. Assuming f ∈ L 2 (Ω) , the variational formulation for this problem writes as follows: we look for u ∈ H 1 0 (Ω), such that for all w ∈ H 1 0 (Ω)

We apply the rst approach developed in Section 2 to derive approximate models of the rst class (Sections A.1.2 and A.1.3) and the second approach developed in Section 3 to derive approximate models of the second class (Sections A.1.4 and A.1.5) for this conguration. Inria where σ 0 > 0 is a given constant. We assume the right-hand side f is a piecewise smooth function that is independent of ε and vanishes inside the layer.

We assume that we have a solution u ∈ H 1 (Ω) to (76). Then, denoting the solution u by

where ∂ n represents the normal derivative in the direction of the normal vector, which is interior to Ω ε ext on Γ ε ext , and exterior to Ω ε int on Γ ε int , as shown at Figure 14. Due to the cylindrical shape of the considered domain, we consider these equations to be written in cylindrical coordinates. Thus, the Laplacian operators of equation (77) have the following form

A key point for the derivation of a multiscale expansion for the solution to Problem (77) consists in performing a scaling along the direction normal to the thin layer. We begin by describing domain Ω ε lay in the following way

where γ is a parametrization of the interface Γ (see Figure 14), which in cylindrical coordinates is dened as γ(θ, z) = (r 0 , θ, z), for all θ ∈ [0, 2π), z ∈ (0, z 0 ), and n = (1, 0, 0) is the normal vector to the curve Γ. This domain geometry induces the following scaling

Inria

As a consequence, we have

This scaling allows us to write the scalar operator

Now we perform an expansion of the terms 1 r 0 + εR and 1 (r 0 + εR) 2 in powers of ε so that we obtain the following expression

We also notice that on the interfaces Γ ε int and Γ ε ext we rewrite the normal derivative in the following form

Finally we denote by U the function that satises

We rewrite Equations (77) with the newly dened variables and functions and they satisfy the following equations outside the thin layer

the following equation inside the thin layer

and the following transmission and boundary conditions

First class of ITCs: construction of a multiscale expansion

We now derive the asymptotic expansion. To begin with, we perform an Ansatz in the form of power series of ε for the solution to Problems (78), ( 79) and (80). We look for solutions

Equations for the coecients of the electric potential Substituting the previous expressions into Equations ( 78), (79), and (80), and collecting the terms with the same powers in ε, for every k ∈ N, we obtain the following set of equations outside the layer

and the following equations inside the layer

along with the following transmission conditions

and the following boundary conditions Inria where

Solving this dierential equation we obtain function ϕ k 0 and thus, the complete expression of U k .

Third step:

We derive the equations outside the layer by employing Equations (82a), (82b), (84a), (84b), (85a), and (85b). We infer that u k int and u k ext are dened independently in the two subdomains Ω ε int and Ω ε ext .

We will now employ this algorithm to obtain the equations for the rst terms of the expansion.

First terms of the asymptotics

Terms of order zero Thanks to the preliminary steps formerly performed during the initialization of the algorithm we already know that U 0 has the form U 0 (R, θ, z) = ϕ 0 0 (θ, z). In the same way we consider Problem (87

We deduce that the solution to this equation has the form U 1 (R, θ, z) = ϕ 1 0 (θ, z). Then, we

We deduce that u 1 ≡ 0.

Terms of order two

We consider Problem (87

We deduce that the solution to this equation has the form U 3 (R, θ, z) = ϕ 3 0 (θ, z). Then, we employ (88) and ϕ 2 0 satises

Solving this problem we obtain the function ϕ 2 0 (θ, z) and thus, the complete expression of U 2 (R, θ, z). Finally, employing (89) we write the problem satised by u 2 outside the layer as two uncoupled problems

Terms of order three

We consider Problem (87) for

RR n°8998

We deduce that the solution to this equation has the form

Then, we employ (88) and we build the following problem for

Solving this problem we obtain the function ϕ 2 0 (θ, z), and thus, the complete expression of U 2 (R, θ, z). Finally, employing (89) we write the problem satised outside the layer by u 3 as two uncoupled problems

Recapitulation of the asymptotic expansion Proposition 6. The asymptotic expansion (81), has the following form

where functions u 0 , u 2 , u 3 , ϕ 2 0 , and ϕ 3 0 are dened by Equations ( 90), ( 93), ( 95), (92), and (94) respectively. For k ∈ N, the solution U k to Equation (87) has the following form

Inria

Proof. We conduct the proof by induction on k. For k = 0, 1, 2, 3, we have already calculated the expressions of u k and U k in the previous section. Now let us assume that for any number i ∈ N, such that i < k, function U i has the form

We begin by considering Problem (87) for U k . Solving this problem we obtain a solution of the form

In the above expression of U k we nd function V k , dened as

at the rst step of the algorithm.

A.2.3 First class of ITCs: equivalent models

Now that we know the expressions for the rst terms of the expansion, we truncate the series and we identify a simpler problem satised by

ext up to a residual term of order ε k+1 . We neglect the residual term of order ε k+1 to obtain an approximate model satised by function u [k] . We formally derive three approximate models of second, third, and fourth order respectively.

Second-order model

For deriving the model of order two, we truncate the series from the second term and we dene u (1) as

From (90), we deduce that u (1) solves the following uncoupled problem

In this case, we have u [1] = u (1) as u (1) does not depend on ε. We infer a second-order model satised by u [1] solution to Problem (96).

Third-order model RR n°8998

For deriving the model of order three, we truncate the series from the third term and we dene u (2) as

From (90), (91), and (93) we deduce that u (2) satises the following equations

where

Following the same procedure as in Section 2.2 we obtain the following third-order asymptotic model for u [2]

Fourth-order model For deriving the model of order four, we truncate the series from the fourth term and we dene u (3) as

From (90), ( 91), (93), and (95), we deduce that u (3) satises the following equations

Following the same procedure as in Section 2.2 we obtain the following fourth-order asymptotic model for u [3] 

Γ ε , u [3] = 0 on ∂Ω ∩ ∂Ω ε .

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A.2.4 Second class of ITCs: construction of a multiscale expansion

In this section we show the asymptotic models we obtain when we write the asymptotic conditions across the interface Γ. We expand the solution in power series of ε. Then, by truncating this series and neglecting higher order terms in ε, we derive approximate models coupled with equivalent transmission conditions across interface Γ. Since we use the same procedure as in the previous sections, we will concentrate on presenting the obtained results, regarding the multiscale expansion and the derivation of the asymptotic models. The domain where the approximate models are dened is depicted at Figure 15. For deriving the equivalent models, we rst use an Ansatz in the form of power series of ε for the solution to problems (78) and (80). We look for solutions

where functions u k int k∈N and u k ext k∈N are now dened in ε-independent domains, contrary to the rst approach. We emphasize that the sequence u k int k∈N (respectively u k ext k∈N ) is dened in Ω int (respectively Ω ext ) even if its associated series does not approach u in the thin layer. We assume that for k ∈ N, the terms u k int and u k ext are as regular as necessary, we refer to [START_REF] Durué | Time-harmonic Maxwell equations in biological cells. The dierential form formalism to treat the thin layer[END_REF] which provides some regularity results. Then, we conduct a formal Taylor series expansion of the terms u k int Γ ε int and u k ext Γ ε ext of the series, in order to write the transmission conditions across interface Γ. We perform the formal Taylor series expansion in the same way as formerly done in Section 3.1. Substituting this Ansatz and the formal Taylor series expansions into Equations (78) and (80), and grouping the terms with the same powers in ε together, for every k ∈ N, we obtain the following set of equations in Ω int and

and the following equations inside the layer

along with the following transmission conditions

and the following boundary conditions

Employing these equations ((100) -( 103)), we determine the elementary problems satised outside and inside the layer for any k ∈ N.

Proposition 7. Following a similar procedure as for the rst class, we deduce that U 0 ≡ U 1 ≡ 0. Thus, the asymptotic expansion (99), has the following form where the functions u 0 and u 1 satisfy the following problems

and

A.2.5 Second class of ITCs: equivalent models Once we know the expressions for the rst terms of the expansion, we truncate the series and we identify a simpler problem satised by u (k) = u 0 + εu 1 + . . . + ε k u k in Ω int ∪ Ω ext up to a residual term of order ε k+1 . We neglect the residual term of order ε k+1 to obtain an approximate model satised by function u [k] . Here, we formally derive two approximate models of order one and order two respectively.

First-order model

Second-order model

int on Γ, u [START_REF] Bendali | The eect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation[END_REF] int = 0 on ∂Ω ∩ ∂Ω int .

ext on Γ, u [START_REF] Bendali | The eect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation[END_REF] ext = 0 on ∂Ω ∩ ∂Ω ext .

(107) RR n°8998