Error estimates for a Neumann problem in highly oscillating thin domains

In this work we analyze convergence of solutions for the Laplace operator with Neumann boundary conditions in a two-dimensional highly oscillating domain which degenerates into a segment (thin domains) of the real line. We consider the case where the height of the thin domain, amplitude and period of the oscillations are all of the same order, given by a small parameter $\epsilon$. We investigate strong convergence properties of the solutions using an appropriate corrector approach. We also give error estimates when we replace the original solutions for the second-order expansion through the Multiple-Scale Method.


Introduction
Several important problems arising in physics and engineering lead to consider boundaryvalue problems in domains with oscillating boundaries, such as flows over rough walls, electromagnetic waves in a region containing a rough interface [20], or heat transmission in winglets [2].
In the present paper we consider the Poisson equation with Neumann boundary conditions, arising from the study of stead state of reaction-diffusion flow over a thin bar with a rough boundary [4]. The aim of our study is to derive precise estimates for the effective behavior of the solutions of such problem.
In order to setup the problem, let g ∈ C 1 (R, R) be a L-periodic positive function. Given a small parameter > 0, let us consider the family of two-dimensional domains R = {(x 1 , x 2 ) ∈ R 2 | 0 < x 1 < 1, 0 < x 2 < g(x 1 / )}. (1.1) We restrict our attention to the solutions w of the family of elliptic equations 2) where N denotes the unit outward normal vector field to ∂R and f is a non-homogeneous term in L 2 (R ) uniformly bounded. Many authors have devoted efforts in the investigation of the asymptotic behavior of a similar equation in related domains, e.g. [2,8,13,19,20]. The basic assumption is that the scale of wavelength of the roughness is small compared with the roughness amplitude. Therefore it is important to emphasize that the amplitude and period of the oscillations of R are of the same order , which also coincides with the order of thickness of the thin domain. This scaling makes the problem very resonant and the determination of the limiting problem ( = 0) is not straightforward. It makes our analysis and results different from those papers.
Going back to problem (1.2), in [4] the authors combine methods from homogenization theory, specially those related to reticulated structures, to obtain the limiting behavior of the family of solutions w . We recall that the convergence obtained in [4] is with respect to the weak topology of the Sobolev space H 1 . As pointed in [3], this convergence is the best one in these spaces. In general, one can not expect strong convergence in homogenization related problems. We refer the reader to [7,11,25,27] for a general introduction to the theory of homogenization and to [12] for a general treatise on reticulated structures.
In such problems, where the analytical solution demands a substantial effort or is unknown, it is of interest in discussing error estimates (in norm) when we replace it with numerical approximations.
In this work we made use of an appropriate corrector approach developed by Bensoussan, Lions and Papanicolaou in [7] to derive a kind of strong convergence and error estimates in H 1 -norm. This can be summarized as: Suppose w w 0 weakly in H 1 (R ) in a sense to be defined. It is possible to find an explicit corrector term for > 0 small enough. The reader is invited to [10] for a classical introduction to correctors approach, [17,18,21] for further discussion on approximations of arbitrary order for related problems, and [9,27] for recent works in this subject as well.
Still related with the limiting behavior of problems in thin domains but in connection with continuity properties of global attractors associated to parabolic problems, we can cite [14,15,22,23,24,26] and their references.
Finally, we would like to observe that although we just treat the Neumann problem for the Poisson equation, we may also consider different conditions in the lateral boundaries of the thin domain R , while preserving the Neumann type boundary condition in the upper and lower boundary. The limiting problem preserves the boundary condition. Note that Dirichlet boundary condition in the upper and lower boundaries is a trivial case since the trace operator is continuous on the set (0, 1) ⊂ ∂R . This paper is organized as follows. In Section 2, we introduce the functional setting for the perturbed problem (1.2) and the limiting one. In Section 3, we provide formal computations to derive the homogenized problem by the Multiple-Scale Method. In Section 4, we obtain convergence in H 1 -norms for the first-order corrector, and in Section 5, we use the second-order corrector to obtain rates of convergence for the first order approximation. We emphasize the relevance of these rates for numerical approximations of solutions of partial differential equations in highly heterogeneous and porous media.

Preliminaries
We stress for the fact that R varies in accordance with a positive parameter and, when goes to 0, the domains R collapse themselves to the interval (0, 1). Therefore, in order to preserve the "relative capacity" of a mensurable subset O ⊂ R , we rescale the Lebesgue measure by 1/ , dealing with the singular measure This measure has been considered in studies involving thin domains, e.g. [15,22,24], and allows us introduce the Lebesgue L 2 (R ; ρ ) and the Sobolev H 1 (R ; ρ ) spaces. The norms in these spaces will be denoted by ||| · ||| L 2 (R ) and ||| · ||| H 1 (R ) respectively, being induced by the inner products respectively.
Remark 2.1. Notice that the ||| · ||| -norms and the usual ones in L 2 (R ) and H 1 (R ) are equivalents and easily related by The variational formulation of (1.2) is: find w ∈ H 1 (R ) such that which is equivalent to find w ∈ H 1 (R ; ρ ) such that Observe that the solutions w satisfy a priori estimates uniformly with respect to . Thus, we can take ϕ = w in (2.1) and (2.2) to obtain In order to capture the limiting behavior of a (w , w ) as → 0, we consider the sesquilinear form a 0 in H 1 (0, 1) given by ds is the average of function g on the interval (0, L). We also will consider L 2 (0, 1) endowed with the norm induced by the inner product (·, ·) 0 , given by (2.5)

Multiscale Expansion of Solutions
In this section we proceed as in [4,7,10,12,25]. We suppose for a moment f (x 1 , x 2 ) = f (x 1 ) for all > 0 and (x 1 , x 2 ) ∈ R . Then we use the Multiple-Scale Method to introduce formally the homogenized equation and the second-order expansion to the solutions w ∈ H 1 (R ) of problem (1.2). We seek for a formal asymptotic expansion of the form The variables x 1 and x 2 represent the "macroscopic" scale on the model, while x 1 / and x 2 / represent the "microscopic" effect of the oscillating phenomena on the thin domain. The fact that R degenerates to a line segment when goes to 0 suggests that the solutions w will not depend on the "macroscopic" variable x 2 . This is taken into account in the choice of w i assuming that w i does not depend on the macroscopic variable x 2 .
In order to construct the functions w i of (3.1), motivated by the periodic nature of R , we consider the basic cell We decompose ∂Y * in the lateral part of the boundary ) ∈ R 2 | 0 < y < L} and the lower part of the boundary B 2 = {(y, 0)) ∈ R 2 | 0 < y < L}. We also assume that Observing from the geometry of R that the unit outward normal vectors to ∂R and to ∂Y * , N = (N 1 , N 2 ) and N = (N 1 , N 2 ) respectively are related by one can see by replacing the expansion (3.1) in the problem (1.2), after equate powers of , that the differential equations for the functions w 0 , w 1 and w 2 are as follows: which implies that w 0 has to be constant with respect to variables y, z, ie, Concerning w 1 , we have Since ∂ x ∂ y w 0 = 0, denoting by X(y, z) the unique solution (up to a constant) of we get For w 2 , the boundary value problem is or, equivalently, upon using the expression for w 1 in (3.8), Remark 3.1. These problems are second order partial differential equations in the variables (y, z) ∈ Y * , with x ∈ (0, 1) playing the role of a parameter.
Next, by Fredholm alternative we can derive the limiting problem for On the other hand, by the Divergence's theorem and (3.9), Therefore replacing this expression in (3.10), we reach Since X depends only on y and z, we can conclude that w 0 must satisfy Now we use the homogenized equation (3.12) to describe w 2 . We rewrite problem (3.9) as 14) The linearity of (3.14) together with the fact that d 2 w 0 dx 2 does not depend on the variables y and z suggest that we look for w 2 (x, y, z) of the form where θ is the solution of the auxiliary problem Then, we can use (3.8) and (3.15) to introduce the following asymptotic expansion for (1.2): This expansion will play an essential role in Section 5 below.
Remark 3.3. According to Bensoussan, Lions and Papanicolaou in [7], the functions X and θ define the first-order correctors and the second-order correctors The functions X and θ are originally defined in the representative cell Y * , but to consider these functions in the thin domain R , we use their periodicities at y to extend them to the band Y = {(y, z) ∈ R 2 | y ∈ R, 0 < z < g(y)}, and we compose them with the diffeomorphisms In the analysis below, with some abuse of notation we will denote these compositions by X (x 1 / , x 2 / ) and θ (x 1 / , x 2 / ) everywhere for (x 1 , x 2 ) ∈ R . With these considerations we can obtain some estimates on R for X and θ as well. It is easy to see that (3.20) Similarly we can get Indeed, the following quantity Remark 3.6. Also, we can use the elliptic form a Y * to show that the homogenized coefficient r is positive. We will perform this here for reader's convenience. For all φ ∈ V , we have that the solution X satisfies Recall that B 1 is the upper part of the boundary of the basic cell. Consequently, y 1 − X satisfies for all φ ∈ V . Also, we have by (3.13) Hence, due to relation (3.22) with φ = −X and identity (3.23), we get

First-order Corrector
As already noted, the solutions w of (1.2) actually do not converge in H 1 -norms. However, if we "improve" w by its first-order corrector, we are able to show the following result 2) with f ∈ L 2 (R ) satisfying |||f ||| L 2 (R ) ≤ C for some C > 0 independent of . Consider the family of functionsf ∈ L 2 (0, 1) defined bŷ where κ is the first-order corrector of w defined in Remark 3.3 and w 0 ∈ H 2 (0, 1) ∩ C 1 (0, 1) is the unique solution of the homogenized equation (3.12) with Proof. By variational formulation of (1.2), we have Thus, observing that w 0 + κ ∈ H 1 (R ) * , we obtain by symmetry of a Using the change of variables (x, y) → (x, y/ ) on [4,Theorem 4.3], it is easy to see that (4.5) * Here w 0 is considered as a function of x 1 and x 2 , simply with some abuse writing w 0 (x 1 , x 2 ) = w 0 (x 1 ).

Second-order corrector
Let w 0 be the homogenized solution (3.12), X and θ be the auxiliary solutions given by (3.7) and (3.16) on the basic cell Y * , which were conveniently defined in the thin domain R by way off Remark 3.4.
In this section, we use the second-order corrector (3.19) to present an error estimate when we replace the solutions w of (1.2) by the first-order truncation with respect to the norm ||| · ||| H 1 (R ) introduced in Section 2. Thus, let us consider the second-order truncation Theorem 5.1. Let R be the thin domain defined in (1.1) and let w be the solution of Then, if W 1 and W 2 are given by (5.1) and (5.2) respectively, we obtain that Consequently, we obtain the following rate for the first-order approximation where K 1 and K 2 are positive constants independent of > 0.
Proof. First, we note that (5.4) is a direct consequence from (5.3) and (3.21), since that for some constant K 2 > 0 independent of . Now we estimate the norm |||φ ||| 2 H 1 (R ) = a (φ , φ ) of the function φ given by In order to do it, we compute a (φ , ϕ) for arbitrary test functions ϕ ∈ H 1 (R ) and establish an estimate of the form Since a is an elliptic form, we can take ϕ = φ in the above inequality to obtain the desired estimate. Using the notation from Section 3, we get by (3.3) that Hence, due to (3.7) and (3.16), we have Consequently, it follows from (1.2) and (3.12) (after some calculations) that That is, in R for all > 0.
On the boundary ∂R , we have by the identity (3.4) and boundary conditions from (3.7) and (3.16) that Thus, the function φ satisfies the following boundary value problem and We consider now the variational formulation of problem (5.5): find φ ∈ H 1 (R ) such that Observe that the function φ must satisfy an uniform a priori estimate on . Indeed, if we take ϕ = φ in the expression (5.8), we obtain We need to get sharp inequalities on F and H to estimate a (φ , φ ). It is clear from their definitions that these estimates will be consequence of those ones for w 0 , X and θ. Since f is a smooth function, we have by classical regularity results given in [1] that the solution w 0 of the homogenized problem is smooth enough to guarantee that its derivatives up to the fourth order are in L ∞ (0, 1). Note that similar statements are also true for X and θ ∈ H 1 (Y * ).
The proof of (5.13) can be found in [12,16]. We recall it here for the reader's convenience. From smoothness of ∂Y * we can define a smooth extension, M = (M 1 , M 2 ) ∈ C 1 (Y * ), of the unitary normal vector field N on Y * , such that M (y, z) = N (y, z) a.e. ∂Y * , and with support of M in a some neighborhood of ∂Y * . Hence, for all ϕ ∈ H 1 (R ) it follows that

Final conclusion
In this work, we give a precise rate of the convergence for solutions of an elliptic problem posed in a family of rough domains with a singular collapsing structure. In our analysis we use the corrector approach grounded in a formal asymptotic expansion of solutions, widely used in homogenization theory, to obtain a rigorous strong convergence result in Theorem 4.1. The second-order corrector is used to obtain an error estimate for the convergence result in Theorem 5.1. We observe that the rate of our convergence result is sharp in the sense that we do not have introduced any boundary layer terms in the correctors, see e.g. [21].