Approximate transmission conditions for a Poisson problem at mid-diffusion

This work consists in the asymptotic analysis of the solution of Poisson equation in a bounded domain of $\mathbb{R}^{P}$ $(P=2,3)$ with a thin layer. We use a method based on hierarchical variational equations to derive asymptotic expansion of the solution with respect to the thickness of the thin layer. We determine the first three terms of the expansion and prove the error estimate made by truncating the expansion after a finite number of terms. Next, using the first two terms of the asymptotic expansion, we show that we can model the effect of the thin layer by a problem with transmission conditions of order two.


Introduction
This paper deals with the study of the asymptotic behavior of the solution of Poisson equation in a bounded domain Ω of R P (P = 2, 3) consisting of two sub-domains separated by a thin layer of thickness δ (destined to tend to 0). The mesh of these thin geometries presents numerical instabilities that can severely damage the accuracy of the entire process of resolution. To overcome this difficulty, we adopt asymptotic methods to model the effect of the thin layer by problems with either appropriate boundary conditions when we consider a domain surrounded by a thin layer (see for instance [2,4,3,10,11]) or, as in this paper, with suitable transmission conditions on the interface (see for instance [6,8,15,16,18,19]). Although this type of conditions has been widely studied, there is still a lot to be understood concerning the effects of thin shell and their modelisation. Our motivation comes from [17,18], in which the authors have worked on problems of electromagnetic and biological origins. We cite for example that of Poignard [17,Chapter 2]. He considered a cell immersed in an ambient medium and studied the electric field in the transverse magnetic (TM) mode at mid-frequency and from which our problem was inspired.
The main result of this paper is to approximate the solution u δ of Problem (1) by a solution of a problem involving Poisson equation in Ω with two sub-domains separated by an arbitrary interface Γ between Γ δ,1 and Γ δ,2 (see Fig. 2 and Fig. 3), with transmission conditions of order two on Γ, modeling the effect of the thin layer. However, it seems that the existence and uniqueness of the solution of this problem are not obvious. Therefore, we rewrite the problem into a pseudodifferential equation (cf. [5]) and show that in the case of mid-diffusion, we can find the appropriate position of the surface Γ to solve this equation. The cases 3D and 2D are similar. We treat the three-dimensional case and the two dimensional one comes as a remark.
The present paper is organized as follows. In Section 2, we give the statement of the model problem considered. In section 3, we collect basic results of differential geometry of surfaces. Sections 4 and 5 are devoted to the asymptotic analysis of our problem. We present, in section 4, hierarchical variational equations suited to the construction of a formal asymptotic expansion up to any order, while Section 5 focuses on the convergence of this ansatz. With the help of the asymptotic expansion of the solution u δ , we model, in the last section, the effect of the thin layer by a problem with appropriate transmission conditions. We consider a parallel surface Γ to Γ δ,1 and Γ δ,2 dividing Ω δ into two thin layers Ω δ,1 and Ω δ,2 of thickness respectively p 1 δ and p 2 δ, where p 1 and p 2 are nonnegative real numbers satisfying p 1 + p 2 = 1 and such that p 1 and p 2 belong to a small neighborhood of 1/2 (see Fig. 2 and Fig. 3). The term small neighborhood means that the constants p 1 and p 2 are not too close to 1 or 0, in order to avoid having a layer too thin compared to the other because the following analysis does not lend itself to this case. Under the aforementioned assumptions, we investigate in H 1 (Ω) the solution u δ := (u i,δ , u d 1 ,δ , u d 2 ,δ , u e,δ ) of the following problem

Problem setting
with transmission conditions where ∂ n denotes the derivative in the direction of the unit normal vector n to Γ (outer for Ω δ,1 and inner for Ω δ,2 ).

Notations and definitions
The goal of this section is to define and to collect the main features of differential geometry [9] (see also [13]) in order to formulate our problem in a fixed domain (independent of δ) which is a key tool to determine the asymptotic expansion of the solution u δ . In the sequel, Greek indice β takes the values 1 and 2. Let I δ,1 = (−δ, 0) and I δ,2 = (0, δ) . We parameterize the thin shell Ω δ,β by the manifold Γ × I δ,β through the mapping ψ β defined by As well-known [9], if the thickness of Ω δ,β is small enough, ψ β is a C ∞ -diffeomorphism of manifolds and it is also known [15, Remark 2.1] that the normal vector n δ,β to Γ δ,β can be identified to n. To each function v β defined on Ω δ,β , we associate the function v β defined on then, we have where ∇ Γ v β (m) and R are respectively the surfacic gradient of v at m ∈ Γ and the curvature operator R of Γ at point m. The volume element on the thin shell Ω δ,β is given by Now, we introduce the scaling s β = η β /δ, and the intervals I 1 = (−1, 0) and I 2 = (0, 1) such that the C ∞ -diffeomorphism Φ β , defined by parameterizes the thin shell Ω δ,β . To any function v β defined on Ω δ,β , we associate the function then the gradient takes the form The volume element on the thin shell Ω δ,β becomes where J δ,β := I + p β δs β R.
Let u β and v β be two regular functions defined on Ω δ,β . From (3) and (4), we get the change of variables formula Remark 1 In the two-dimensional case, if m ∈ Γ, we parameterize the curve Γ by m(t) where t ∈ (0, l Γ ) is the curvilinear abscissa and l Γ is the length of the curve Γ, then formula (5) turns into Remark 2 For any function u defined in a neighborhood of Γ, we denote, for convenience, by u |Γ the trace of u on Γ indifferently in local coordinates or in Cartesian coordinates.

The asymptotic analysis
This section is devoted to the asymptotic analysis of the solution of Problem (2). We show that this latter is equivalent to a variational equation from which we derive the asymptotic expansion of u δ . We give a hierarchy of variational equations needed to determine the terms of the expansion and we calculate the first two terms of the expansion.

Calculation of the first terms
In this paragraph, we first recall some theoretical results needed for our calculation. After, we calculate explicitly the first two terms of Expansions (10)- (11) and (13) in order to present a recursive method to define successively the terms of these expansions. Let s be a nonnegative real number. We define P H s (Ω) (see [14]) the space of functions in Ω, with H s -regularity in Ω e and Ω i , as follows We need the following theorem. Its proof [17, p. 122] is an application of the reflection principle [12, p. 147].
Theorem 4 Let G belongs to H s (Γ), s ≥ −1/2. Then the following problem in Ω e , admits a unique solution U = (U i , U e ) in P H s+3/2 (Ω). Moreover, if m is a nonnegative integer, and s > m + P − 1 2 . Then U i ∈ C m (Ω i ) and U e ∈ C m (Ω e ).
We also need the following technical lemma. This construction was motivated by [4] and its proof is a straightforward verification.
Lemma 5 For β = 1, 2, let q [β] be a given function in L 2 (Γ) and let k [β] be a vectorial function in L 2 (Ω β , C 3 ) such that the partial application s β → k [β] (., s β ) is valued in the space of vectorial fields tangent to Γ and also div Γ k [β] ∈ L 2 (Ω β ). Then the solution h [β] of the variational equation is explicitly given by
Elliptic regularity results (see e.g. [1]) show that if f belongs to C ∞ (Ω), then (u i,0 , u e,0 ) is a well-defined element of C ∞ (Ω i ) × C ∞ (Ω e ). As a consequence the first term is determined.

Convergence Theorem
The process described in the previous section can be continued up to any order provided that the data are sufficiently regular. We can also estimate the error made by truncating the series after a finite number of terms. Let n be in N, we set j (m, s β ); ∀x = Φ β (m, s β ) ∈ Ω δ,β .
Theorem 6 (Convergence Theorem) For all integers n, there exists a constant c independent of δ such as Proof. Since f is C ∞ , all terms in Expansions (10), (11) and (13) up to order n + 1 may be obtained from Equations (22)-(26). Let us define the remainders R D 1 ,n , R D 2 ,n , R 1,n and R 2,n of Taylor expansions in the normal variable with respect to δ up to order n of u where s β ∈ I β . We shall rely on the following proposition to show the estimates of the remainders R D β ,n and R β,n . The steps of the proof are very similar to those given in [18,Section 5]. We refer the reader to this paper.
Now, we use the fact that u By the estimates based on the explicit expressions of the bilinear form a k,l (., .) and those of Propositions 7, we have L 2 (Ω β ) Since δ is small enough, we have, ∀v ∈ H 1 0 (Ω), .

Approximate transmission conditions
This section is devoted to the approximation of u δ by a solution of a problem modelling the effect of the thin layer with a precision of order two in δ. We truncate the series defining the asymptotic expansions, keeping only the first two terms i,δ = u i,0 + δu i,1 in Ω i , u e,δ ≃ u (1) e,δ = u e,0 + δu e, 1 in Ω e , u d 1 ,δ (x) ≃ u (1) d 1 ,δ (m, s 1 ) = u [1] 0 (m, s 1 ) + δu [1] 1 (m, s 1 ), ∀x = Φ 1 (m, s 1 ) ∈ Ω δ,1 , u d 2 ,δ (x) ≃ u (1) d 2 ,δ (m, s 2 ) := u [2] 0 (m, s 2 ) + δu [2] 1 (m, s 2 ), ∀x = Φ 2 (m, e,δ|Γ = δB u with Let U ap δ := u ap i,δ , u ap e,δ be the solution of (40) with ρ δ = 0 and ξ δ = 0. We obtain a problem (P ap δ ) with transmission conditions of order equal to that of the differential operator. The new transmission conditions on Γ are defined by However, the bilinear form associated to Problem (P ap δ ) is neither positive nor negative. Then the existence and uniqueness of the solution are not ensured by the Lax-Milgram lemma. Therefore, we reformulate Problem (P ap δ ) into a nonlocal equation on the interface Γ (cf. [5]). A direct use of transmission conditions (41) leads to an operator which is not self-adjoint. So, we choose the position of Γ in such a way that the jump of the trace of the solution on Γ is null. We put which is valid only when α i < α δ < α e or α e < α δ < α i . This corresponds to the case of mid-diffusion. Transmission conditions (41) become    u ap i,δ|Γ − u ap e,δ|Γ = 0, α i ∂ n u ap i,δ|Γ − α e ∂ n u ap e,δ|Γ = δ After, we remove the right-hand side of Problem (P ap δ ) by a standard lift: let G be in H 1 0 (Ω) such that −div (α∇G) = f. Then the function Ψ = U ap δ − G solves the following problem in Ω e , Ψ i|Γ − Ψ e|Γ = 0 on Γ, We introduce the Steklov-Poicaré operators S i and S e (called also Dirichlet-to-Neumann operators) defined from H 1/2 (Γ) onto H −1/2 (Γ) by S i ϕ := ∂ n u i|Γ , where u i is the solution of the boundary value problem −∆u i = 0 in Ω i , u i|Γ = ϕ on Γ, and by S e ψ := ∂ −n u e|Γ , where u e is the solution of the boundary value problem    −∆u e = 0 in Ω e , u e|Γ = ψ on Γ, u e|∂Ω = 0 on ∂Ω.
Then (P ap δ ) is equivalent to the boundary equation where ω is the trace of Ψ on the surface Γ. We are now in position to state the existence and uniqueness theorem, which proof is similar to that of Theorem 2.5 in [5].
We can now formulate our main result.