Two-scale homogenization of the Poisson equation with friction boundary condition in a perforated domain

Abstract

We study the homogenization of the Poisson equation in a periodically perforated domain, of period $ε > 0$ , with a friction type boundary condition on the holes’ boundary. This non-linear condition allows the solution to be non-zero on the periodic boundary if some conditions are satisfied. Using two-scale convergence results we prove that the solution of the mixed variational formulation converges, as ε goes to 0, to the solution of a two-scale mixed problem. We also prove that this homogenized problem is well-posed. A numerical test is done, using the Finite Element Method and a quadratic programming algorithm, in order to compare the heterogeneous and homogenized solutions.

Keywords

homogenization variational inequalities mixed formulation

1. Introduction

In this paper, we study the homogenization of the Poisson equation posed in a periodically perforated domain, of period $ε > 0$ and holes of order ε. On the holes’ boundary we impose a friction type boundary condition whereas on the external boundary we impose a homogeneous Dirichlet condition. The friction boundary condition states that the absolute value of the normal derivative on the periodic boundary is bounded by a positive threshold function. If the absolute value of the normal derivative reaches the threshold then the solution of the equation is proportional to the normal derivative; else, if the absolute value of the normal derivative is strictly inferior to the threshold function, then the solution is equal to zero on the periodic boundary. Therefore, homogeneous Dirichlet and Robin (or Fourier) boundary conditions are allowed in different and unknown parts of the periodic boundary, depending on the value of the normal derivative. The coefficient in the case of the Robin condition is also unknown.

This problem could model the stationary vertical displacements of a periodically perforated membrane in which, due to the friction boundary condition, the displacements on the holes’ boundary could be either zero or proportional to the normal strain, depending on the values of the late with respect to the threshold function.

The variational formulation of this problem is a variational inequality of the second kind which can be seen as an unconstrained non-differentiable minimization problem (see, for example, [10]). An equivalent formulation can be obtained using a Lagrange multiplier which will relax the non-differentiable problem into a constrained differentiable saddle-point problem. The Euler–Lagrange optimality condition leads to a mixed variational problem. From a mathematical analysis point of view, the study of the variational inequality is enough to prove the well-posedness, i.e., the existence and uniqueness of solutions to the heterogeneous problem for each ε, using classical results (see [9]). However, from the numerical and scientific computing point of view, the mixed variational formulation is more adapted, since we do not have to deal with a non-differentiable term (see [10]) and, after discretization, it leads to the resolution of a constrained quadratic programming problem.

The goal of this work is to study the asymptotic behavior of the solution as ε goes to zero and to find the homogenized problem that the solution’s limit satisfies. To this end, we will use, first, the two-scale asymptotic development method (see [4]) to formally derive a homogenized variational inequality. Next, we will use the two-scale homogenization method (see [1,15]). In particular, two-scale convergence results in [1,3] will be used to prove the convergence of the solution of the heterogeneous mixed problem to the solution of a two-scale homogenized mixed variational problem. We point out that we could study the asymptotic behavior of the solution of the variational inequality of the second kind, using convergences results in [16,17], to obtain in the limit a homogenized variational inequality. However, as we stated before, a homogenized mixed variational problem would be preferred as we intend to do numerical simulations.

There are other works on the homogenization of variational inequalities as for example [6] and more recently [5]. In [6], the authors study the homogenization of a Signorini problem in a perforated domain involving a positivity condition on the boundary of the holes. The authors proved, among other results, that when the size of the hole is of the order of the period, as in our case, then in the limit problem the positivity condition spreads into the whole domain. Moreover, the homogenized coefficient matrix corresponds to the classical homogenized matrix in perforated domains.

In [5], the authors study the homogenization of an elasticity problem in a heterogeneous medium with contact conditions using the periodic unfolding method. In this case, due to the scaling of the Tresca coefficient, the limit problem corresponds to a partial differential equation, whereas the corrector problem is a variational inequality.

The present work has to be seen as a first step toward the study, in a forthcoming paper, of the homogenization of the Stokes problem in a periodically perforated domain with friction type boundary condition on the periodic boundary (see [8]), using the two-scale convergence method and mixed formulations (in view of a numerical implementation).

The outline of the paper is as follows. In Section 2, we set up the problem, that is, we define the perforated domain and we state the Poisson equation with the friction boundary condition. In Section 3, we present the variational formulations of the problem and we state the existence and uniqueness of the solution, in particular for the heterogeneous mixed variational formulation. At the end of this section we introduce the two-scale homogenized mixed problem and we state our main convergence result. In Section 4, we present the homogenization process. Firstly, we use the formal two-scale development method to derive a homogenized variational inequality of the second kind. Next, we obtain a priori estimates for the solution of the mixed heterogeneous problem that allow us to pass to the limit and prove the convergence result stated in Section 3. At the end of Section 4, we prove an inf–sup condition for the two-scale homogenized mixed problem. Finally, in Section 5 we present a numerical test in order to compare the heterogeneous solution and the solution of the homogenized mixed problem.

2. Setting up the problem

Let $Ω \subset R^{2}$ be an open bounded set. Let $Y = {] 0, 1 [}^{2}$ , the unit cell. Let T be an open set of $R^{2}$ , with smooth boundary Γ, such that $\overline{T} \subset Y$ . We denote by $Y^{*}$ the perforated unit cell, i.e., $Y^{*} = Y ∖ \overline{T}$ . Let $ε > 0$ be a small parameter. Let $κ = (k_{1}, k_{2}) \in Z^{2}$ , a multi-index. We denote by $Y_{κ, ε}^{*}$ and $T_{κ, ε}$ the homothetic translation of $Y^{*}$ and T, respectively, that is $\begin{matrix} Y_{κ, ε}^{*} = ε (Y^{*} + κ) = {x \in R^{2} : x = ε (y + κ), y \in Y^{*}} and T_{κ, ε} = ε (T + κ) . \end{matrix}$ Let $I$ be the set of all multi-index κ such that $T_{κ, ε}$ is strictly contained in Ω, that is $I = {κ \in Z^{2} : \overline{T_{κ, ε}} \subset Ω}$ .

Finally we define the perforated domain $Ω^{ε}$ as $\begin{matrix} Ω^{ε} = Ω ∖ ⋃_{κ \in I} \overline{T_{κ, ε}} . \end{matrix}$ The boundary of $Ω^{ε}$ , denoted by $\partial Ω^{ε}$ , is composed of two parts, the external boundary $\partial Ω$ (the boundary of Ω) and the internal boundary $Γ^{ε}$ defined by $\begin{matrix} Γ^{ε} = ⋃_{κ \in I} Γ_{κ, ε}, \end{matrix}$ where $Γ_{κ, ε}$ denotes the boundary of $T_{κ, ε}$ .

We consider the following elliptic problem with a friction type boundary condition (see [10,14]).

Find $u^{ε} : Ω^{ε} \to R$ such that $\begin{array}{rcl} (1) & - Δ u^{ε} = f in Ω^{ε}, \\ (2) & u^{ε} = 0 on \partial Ω, \\ (3) & | \frac{\partial u^{ε}}{\partial n} | ⩽ g^{ε}, \frac{\partial u^{ε}}{\partial n} u^{ε} + g^{ε} | u^{ε} | = 0 on Γ^{ε}, \end{array}$ where $f \in L^{2} (Ω)$ and $g^{ε} : Γ^{ε} \to R$ is the threshold function defined by $g^{ε} (x) = ε g (x / ε)$ a.e. $x \in Γ^{ε}$ , with $g \in L^{2} (Γ)$ , Y-periodic and $g > 0$ a.e. on Γ. The second condition in (3) will be called complementary condition. Condition (3) is equivalent to the existence of a positive function $x \in Γ^{ε} \mapsto α^{ε} (x)$ such that $\begin{matrix} u^{ε} (x) = \{\begin{matrix} 0 & if | \frac{\partial u^{ε}}{\partial n} (x) | < g^{ε} (x), \\ - α^{ε} (x) \frac{\partial u^{ε}}{\partial n} (x) & if | \frac{\partial u^{ε}}{\partial n} (x) | = g^{ε} (x), \end{matrix} x \in Γ^{ε} . \end{matrix}$ The boundary condition (3) is also equivalent to $- \frac{\partial u^{ε}}{\partial n} \in \partial (| g^{ε} |)$ , where $\partial (\cdot)$ denotes the subdifferential of a convex function (see [7]).

3. Variational formulations and main result

3.1. Variational inequality and minimization problem

Let $V^{ε} = {v \in H^{1} (Ω^{ε}) : v = 0 on \partial Ω}$ . The variational formulation of (1)–(3) is the following variational inequality of the second kind (see [10]).

Find $u^{ε} \in V^{ε}$ such that $\begin{matrix} (4) & a^{ε} (u^{ε}, v^{ε} - u^{ε}) + j^{ε} (v^{ε}) - j^{ε} (u^{ε}) ⩾ ℓ^{ε} (v^{ε} - u^{ε}) \forall v^{ε} \in V^{ε}, \end{matrix}$ with $\begin{matrix} a^{ε} (u^{ε}, v^{ε}) = \int_{Ω^{ε}} \nabla u^{ε} \cdot \nabla v^{ε} d x, j^{ε} (v^{ε}) = \int_{Γ^{ε}} g^{ε} | v^{ε} |_{Γ^{ε}} | d Γ, ℓ^{ε} (v^{ε}) = \int_{Ω^{ε}} f v^{ε} d x, \end{matrix}$ where $v^{ε} |_{Γ^{ε}}$ stands for the trace of $v^{ε}$ on $Γ^{ε}$ . The variational inequality (4) is equivalent to the following minimization problem.

Find $u^{ε} \in V^{ε}$ such that $\begin{matrix} (5) & J^{ε} (u^{ε}) = inf_{v^{ε} \in V^{ε}} J^{ε} (v^{ε}), \end{matrix}$ where $J^{ε} : V^{ε} \to R$ is the functional defined by $\begin{matrix} J^{ε} (v^{ε}) = \frac{1}{2} a^{ε} (v^{ε}, v^{ε}) - ℓ^{ε} (v^{ε}) + j^{ε} (v^{ε}) . \end{matrix}$ Since $a^{ε}$ is a coercive bilinear form, $ℓ^{ε}$ is a continuous linear form and $j^{ε}$ is a convex, lower semi-continuous and proper functional, then, following [9], the minimization problem (5) has a unique solution $u^{ε} \in V^{ε}$ .

3.2. Mixed formulation

Using a Lagrange multiplier it is possible to transform the non-differentiable minimization problem (5) into a smooth one (see [10]). To this end, we introduce the closed convex set of $H^{- 1 / 2} (Γ^{ε})$ , $\begin{matrix} Q^{ε} = {μ^{ε} \in L^{2} (Γ^{ε}) : | μ^{ε} | ⩽ g^{ε} a.e. on Γ^{ε}}, \end{matrix}$ and the bilinear form $b^{ε} : L^{2} (Γ^{ε}) \times V^{ε} \to R$ defined by $\begin{matrix} b (μ^{ε}, v^{ε}) = \int_{Γ^{ε}} μ^{ε} v^{ε} |_{Γ^{ε}} d Γ = ⟨ μ^{ε}, v^{ε} |_{Γ^{ε}} ⟩, \end{matrix}$ where $⟨ \cdot, \cdot ⟩$ stands for the duality product between $H^{- 1 / 2} (Γ^{ε})$ and $H^{1 / 2} (Γ^{ε})$ . Then, we have $\begin{matrix} inf_{v^{ε} \in V^{ε}} J^{ε} (v^{ε}) = inf_{v^{ε} \in V^{ε}} sup_{μ^{ε} \in Q^{ε}} (\frac{1}{2} a^{ε} (v^{ε}, v^{ε}) - ℓ^{ε} (v^{ε}) + b^{ε} (μ^{ε}, v^{ε})), \end{matrix}$ and the saddle point $(u^{ε}, λ^{ε})$ can be characterized as the solution of following mixed variational problem.

Find $(u^{ε}, λ^{ε}) \in V^{ε} \times Q^{ε}$ such that $\begin{array}{rcl} (6) & a^{ε} (u^{ε}, v^{ε}) + b^{ε} (λ^{ε}, v^{ε}) = ℓ^{ε} (v^{ε}) \forall v^{ε} \in V^{ε}, \\ (7) & b^{ε} (μ^{ε} - λ^{ε}, u^{ε}) ⩽ 0 \forall μ^{ε} \in Q^{ε} . \end{array}$

The following inf–sup condition, which warranties the existence and uniqueness of a saddle-point $(u^{ε}, λ^{ε})$ of (6)–(7), holds:

Proposition 3.1.
There exists $β_{ε} > 0$ , such that for all $μ^{ε} \in H^{- 1 / 2} (Γ^{ε})$ $\begin{matrix} sup_{v^{ε} \in V^{ε}} \frac{b (μ^{ε}, v^{ε})}{‖ v^{ε} ‖_{1, Ω^{ε}} ‖ μ^{ε} ‖_{- 1 / 2, Γ^{ε}}} ⩾ β_{ε} . \end{matrix}$
Proof.
Let $μ^{ε} \in H^{- 1 / 2} (Γ^{ε})$ . We consider the inverse of the Riesz map $r : H^{1 / 2} (Γ^{ε}) \to H^{- 1 / 2} (Γ^{ε})$ . Let $m^{ε} = r^{- 1} (μ^{ε})$ then $m^{ε} \in H^{1 / 2} (Γ^{ε})$ and $‖ m^{ε} ‖_{1 / 2, Γ^{ε}} = ‖ μ^{ε} ‖_{- 1 / 2, Γ^{ε}}$ .

Using a harmonic extension of $m^{ε}$ we have that there exists ${\hat{v}}^{ε} \in V^{ε}$ such that $\begin{matrix} {\hat{v}}^{ε} |_{Γ^{ε}} = m^{ε} and {‖ {\hat{v}}^{ε} ‖}_{1, Ω^{ε}} ⩽ C_{1} {‖ m^{ε} ‖}_{1 / 2, Γ^{ε}}, \end{matrix}$ with $C_{1} > 0$ a constant that depends on $Ω^{ε}$ .

Hence, we have $\begin{matrix} b (μ^{ε}, v^{ε}) = ⟨ μ^{ε}, v^{ε} |_{Γ^{ε}} ⟩ = ⟨ μ^{ε}, m^{ε} ⟩ = {‖ μ^{ε} ‖}_{- 1 / 2, Γ^{ε}} {‖ m^{ε} ‖}_{1 / 2, Γ^{ε}} ⩾ \frac{1}{C_{1}} {‖ {\hat{v}}^{ε} ‖}_{1, Ω^{ε}} {‖ μ^{ε} ‖}_{- 1 / 2, Γ^{ε}}, \end{matrix}$ and then the inf–sup condition holds with $β_{ε} = 1 / C_{1} > 0$ . □

In what follows we will use the following weak characterization of $Q^{ε}$ (see [11]) $\begin{matrix} Q^{ε} = {μ^{ε} \in L^{2} (Γ^{ε}) : \int_{Γ^{ε}} μ^{ε} ψ^{ε} d Γ + \int_{Γ^{ε}} g^{ε} | ψ^{ε} | d Γ ⩾ 0, \forall ψ^{ε} \in L^{2} (Γ^{ε})} . \end{matrix}$
3.3. Homogenized two-scale problem and main convergence result

We begin by introducing the functional spaces $V = H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (Y^{*}) / R)$ , with $H_{#}^{1} (Y^{*})$ the space of Y-periodic $H^{1}$ functions, $Y = L^{2} (Ω; L^{2} (Γ))$ and the bilinear and linear forms $\begin{matrix} a_{h} : V \times V \to R, b_{h} : Y \times V \to R, ℓ_{h} : V \to R, \end{matrix}$ respectively defined by $\begin{matrix} \begin{matrix} a_{h} ((u^{0}, u^{1}), (v^{0}, v^{1})) = \int_{Ω} \int_{Y} χ (y) (\nabla u^{0} (x) + \nabla_{y} u^{1} (x, y)) \cdot (\nabla v^{0} (x) + \nabla_{y} v^{1} (x, y)) d y d x, \\ b_{h} (M^{0}, (v^{0}, v^{1})) = \int_{Ω} \int_{Γ} M^{0} (x, y) v^{0} (x) d_{y} Γ d x, \end{matrix} \end{matrix}$ and $\begin{matrix} ℓ_{h} ((v^{0}, v^{1})) = \int_{Ω} \int_{Y} χ (y) f (x) v^{0} (x) d y d x \end{matrix}$ (where χ stands for the characteristic function of $Y^{*}$ and the subscript $_{y}$ denotes differentiation or integration with respect to $y \in Y$ ), for all $(u^{0}, u^{1}), (v^{0}, v^{1}) \in V$ and all $M^{0} \in Y$ . V is endowed with the standard product space norm $‖ (u^{0}, u^{1}) ‖_{1}^{2} = ‖ u^{0} ‖_{1, Ω}^{2} + ‖ u^{1} ‖_{L^{2} (Ω; H_{#}^{1} (Y^{*}))}^{2}$ . Let $Q \subset Y$ be the closed convex set defined by $\begin{matrix} Q = {M^{0} \in Y : \int_{Ω} \int_{Γ} M^{0} (x, y) ψ (x) d_{y} Γ d x + \int_{Ω} \int_{Γ} g (y) | ψ (x) | d_{y} Γ d x ⩾ 0, \forall ψ \in L^{2} (Ω)} . \end{matrix}$

Next, we consider the following two-scale mixed problem.

Find $((u^{0}, u^{1}), L^{0}) \in V \times Q$ such that $\begin{array}{rcl} (8) & a_{h} ((u^{0}, u^{1}), (v^{0}, v^{1})) + b_{h} (L^{0}, (v^{0}, v^{1})) = ℓ_{h} ((v^{0}, v^{1})) \forall (v^{0}, v^{1}) \in V, \\ (9) & b_{h} (M^{0} - L^{0}, (u^{0}, u^{1})) ⩽ 0 \forall M^{0} \in Q . \end{array}$

Remark 3.1.
Note that $L^{0}$ will be unique up to an additive function $c \in L^{2} (Ω; L^{2} (Γ))$ such that $\int_{Γ} c (x, y) d_{y} Γ = 0$ a.e. in Ω, that is, $c \in L^{2} (Ω; L_{0}^{2} (Γ))$ .

In Section 4 we will prove that an inf–sup condition holds for a modified problem that take into account Remark 3.1.

Finally, we state our main convergence result. Theorem 3.2.
Let $g \in C_{#} (Y)$ . Let $(u^{ε}, λ^{ε}) \in V^{ε} \times Q^{ε}$ be the solution of ( 6 )–( 7 ) and let $((u^{0}, u^{1}), L^{0} + c) \in V \times Q$ , $c \in L^{2} (Ω; L_{0}^{2} (Γ))$ , be the solution of ( 8 )–( 9 ). Then, up to a subsequence, the following two-scale convergences hold $\begin{matrix} \tilde{u^{ε}} \overset{2}{⇀} χ (y) u^{0} (x), \tilde{\nabla u^{ε}} \overset{2}{⇀} χ (y) (\nabla u^{0} + \nabla_{y} u^{1} (x, y)) \end{matrix}$ and $\begin{matrix} ε^{- 1} λ^{ε} \overset{2}{⇀} L^{0} (in the sense of (33)) . \end{matrix}$

4. Homogenization process

4.1. Two-scale asymptotic expansion: Formal homogenized variational inequality

In this section we use the two-scale asymptotic expansion method (see, for example, [4]) to formally derive a homogenized problem associated to (1)–(3). The microscopic variable y is defined by $y = x / ε$ .

We propose the following ansatz $\begin{matrix} u^{ε} (x) = u^{0} (x, x / ε) + ε u^{1} (x, x / ε) + ε^{2} u^{2} (x, x / ε) + \dots, \end{matrix}$ where $u^{j} : Ω \times Y \to R$ are regular Y-periodic functions, $j = 0, 1, \dots$ .

As usual we replace in (1)–(3), $u^{ε}$ by its ansatz. The asymptotic expansions of (1) and (2) are standard calculations. We will look in details the asymptotic expansion of the non-linear boundary condition (3). In one hand, from the first condition in (3) we have that $\begin{matrix} (10) & \frac{\partial u^{ε}}{\partial n} ⩽ ε g (\frac{x}{ε}) and \frac{\partial u^{ε}}{\partial n} ⩾ - ε g (\frac{x}{ε}), \end{matrix}$ and, in the other hand, using the development of $u^{ε}$ , that $\begin{matrix} (11) & \frac{\partial u^{ε}}{\partial n} = ε^{- 1} \frac{\partial u^{0}}{\partial_{y} n} + ε^{0} (\frac{\partial u^{0}}{\partial n} + \frac{\partial u^{1}}{\partial_{y} n}) + ε^{1} (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) + \dots . \end{matrix}$ Therefore, comparing orders in both inequalities in (10), we see that they hold if $\begin{array}{rcl} (12) & (ε^{- 1} :) \frac{\partial u^{0}}{\partial_{y} n} = 0, \\ (13) & (ε^{0} :) \frac{\partial u^{0}}{\partial n} + \frac{\partial u^{1}}{\partial_{y} n} = 0, \\ (14) & (ε^{1} :) | \frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n} | ⩽ g (y), \end{array}$ and all the other terms in the development (11) are equal to 0. Next, in the complementary condition of the friction boundary condition (3) we use that $| u^{ε} | = sgn (u^{ε}) u^{ε}$ , and $sgn (u^{ε}) = sgn (u^{0})$ since $u^{0}$ is the leading order in the ansatz (we suppose here that $ε > 0$ is small enough), so $\begin{array}{rcl} 0 & = & \frac{\partial u^{ε}}{\partial n} u^{ε} + g^{ε} | u^{ε} | \\ = & [ε^{- 1} \frac{\partial u^{0}}{\partial_{y} n} + ε^{0} (\frac{\partial u^{0}}{\partial n} + \frac{\partial u^{1}}{\partial_{y} n}) + ε^{1} (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) + \dots] \times (u^{0} + ε u^{1} + ε^{2} u^{2} + \dots) \\ + ε g [sgn (u^{0})] (u^{0} + ε u^{1} + ε^{2} u^{2} + \dots), \end{array}$ and using (12) and (13) we obtain that at order ε $\begin{matrix} (15) & (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) u^{0} + g (y) [sgn (u^{0})] u^{0} = 0, y \in Γ . \end{matrix}$ Finally, comparing orders in the expansion of (1) and (2) and then using (12)–(15), we obtain the following local problems on $Y^{*}$ , for different orders of ε: $\begin{array}{rcl} (16) & - {div}_{y} (\nabla_{y} u^{0}) = 0 in Y^{*}, \\ (17) & \frac{\partial u^{0}}{\partial_{y} n} = 0 on Γ, \\ (18) & u^{0} Y -periodic, \end{array}$ $\begin{array}{rcl} (19) & - {div}_{y} (\nabla u^{0} + \nabla_{y} u^{1}) = 0 in Y^{*}, \\ (20) & \frac{\partial u^{1}}{\partial_{y} n} = - \frac{\partial u^{0}}{\partial n} on Γ, \\ (21) & u^{1} Y -periodic, \end{array}$ $\begin{array}{rcl} (22) & - {div}_{y} (\nabla u^{1} + \nabla_{y} u^{2}) = f + div (\nabla u^{0} + \nabla_{y} u^{1}) in Y^{*}, \\ (23) & | \frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n} | ⩽ g, (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) u^{0} + g | u^{0} | = 0 on Γ, \\ (24) & u^{2} Y -periodic . \end{array}$

From the first local problem (16)–(18), we have that $u^{0}$ only depends on the macroscopic variable x, that is $\begin{matrix} u^{0} (x, y) = u^{0} (x) . \end{matrix}$

The second local problem (19)–(21) will provide us the first corrector $u^{1}$ in terms of $u^{0}$ .

The third local problem (22)–(24) will provide us the homogenized problem in a weak form. First, we multiply Eq. (22) by $u^{0}$ and we integrate by parts in $Y^{*} \times Ω$ . Since $u^{0}$ does not depend on y we obtain $\begin{matrix} - \int_{Ω} \int_{Γ} (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) u^{0} d_{y} Γ d x = \int_{Ω} \int_{Y^{*}} (f + div (\nabla u^{0} + \nabla_{y} u^{1})) u^{0} d y d x . \end{matrix}$ Using the complementary condition in (23) we have $\begin{matrix} (25) & \int_{Ω} \int_{Γ} g | u^{0} | d_{y} Γ d x = \int_{Ω} \int_{Y^{*}} (f + div (\nabla u^{0} + \nabla_{y} u^{1})) u^{0} d y d x . \end{matrix}$ Next, if we multiply (22) by v, a regular enough test function that vanishes on $\partial Ω$ and does not depend on y, we obtain $\begin{matrix} - \int_{Ω} \int_{Γ} (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) v d_{y} Γ d x = \int_{Ω} \int_{Y^{*}} (f + div (\nabla u^{0} + \nabla_{y} u^{1})) v d y d x . \end{matrix}$ Since $\begin{matrix} | \frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n} | ⩽ g (y), a.e. y \in Γ, \end{matrix}$ which is equivalent to $\begin{matrix} \int_{Γ} (\frac{\partial u^{1}}{\partial n} + \frac{\partial u^{2}}{\partial_{y} n}) ψ d_{y} Γ + \int_{Γ} g | ψ | d_{y} Γ ⩾ 0 \forall ψ \in L^{2} (Γ), \end{matrix}$ then if we consider, for each $x \in Ω$ , $ψ (y) = v (x)$ , for all $y \in Γ$ (i.e. ψ constant on Γ) then $\begin{matrix} (26) & \int_{Ω} \int_{Γ} g | v | d_{y} Γ d x ⩾ \int_{Ω} \int_{Y^{*}} (f + div (\nabla u^{0} + \nabla_{y} u^{1})) v d y d x . \end{matrix}$ Subtracting (25) from (26) and integrating by parts in Ω we formally obtain the following variational inequality:

Find $u^{0} \in H_{0}^{1} (Ω)$ such that $\begin{array}{l} \int_{Ω} \int_{Y^{*}} (\nabla u^{0} + \nabla_{y} u^{1}) \cdot \nabla (v - u^{0}) d y d x \\ (27) & + \int_{Ω} \int_{Γ} g | v | d_{y} Γ d x - \int_{Ω} \int_{Γ} g | u^{0} | d_{y} Γ d x ⩾ | Y^{*} | \int_{Ω} f (v - u^{0}) d x \forall v \in H_{0}^{1} (Ω), \end{array}$ where $| \cdot |$ when applied to a set denotes its measure.

Remark 4.1.
A strong homogenized problem could be obtained by integrating in $Y^{}$ Eq. (22), which will give $\int_{Γ} (\partial u^{1} / \partial n + \partial u^{2} / \partial_{y} n) d_{y} Γ$ in terms of $u^{0}$ and $u^{1}$ , and then by integrating on Γ (23). We would obtain that $\begin{matrix} | | Y^{} | f + {div}_{x} (| Y^{} | \nabla u^{0} + \int_{Y^{}} \nabla_{y} u^{1} d y) | ⩽ \int_{Γ} g d_{y} Γ in Ω, \end{matrix}$ and $\begin{matrix} (- | Y^{} | f - {div}_{x} (| Y^{} | \nabla u^{0} + \int_{Y^{*}} \nabla_{y} u^{1} d y)) u^{0} + (\int_{Γ} g d_{y} Γ) | u_{0} | = 0 in Ω . \end{matrix}$

4.2. Proof of the two-scale convergence result

This section is devoted to prove Theorem 3.2. First, we obtain a priori estimates for $u^{ε}$ , independent of ε, and for $λ^{ε}$ , depending on $ε^{1 / 2}$ (see Proposition 4.1). Then, using this result we will pass to the two-scale limit, as $ε \to 0$ , in the mixed problem (6)–(7). Finally, we prove that the two-scale mixed problem (8)–(9) has a unique solution.

4.2.1. A priori estimates

In this part, we obtain a priori estimates for $u^{ε}$ and $λ^{ε}$ . We will prove the following proposition.

Proposition 4.1.
Let $(u^{ε}, λ^{ε})$ be the solution of the mixed variational problem ( 6 )–( 7 ). Then, for all $ε > 0$ , $\begin{array}{rcl} (28) & {‖ u^{ε} ‖}_{1, Ω^{ε}} ⩽ C, \\ (29) & ‖ λ_{ε} ‖_{0, Γ^{ε}} ⩽ C ε^{1 / 2} ‖ g ‖_{0, Γ}, \end{array}$ with C a positive constant that not depends on ε.
Proof.
Since $(u^{ε}, λ^{ε}) \in V^{ε} \times Q^{ε}$ , then $λ^{ε} \in L^{2} (Γ^{ε})$ is such that $\begin{matrix} (30) & \int_{Γ^{ε}} λ^{ε} φ^{ε} d Γ + \int_{Γ^{ε}} g^{ε} | φ^{ε} | d Γ ⩾ 0 \end{matrix}$ for all $φ^{ε} \in L^{2} (Γ^{ε})$ . Using $φ^{ε} = - λ^{ε}$ in (30) we have $\begin{matrix} \int_{Γ^{ε}} {| λ^{ε} |}^{2} d Γ ⩽ \int_{Γ^{ε}} g^{ε} | λ^{ε} | d Γ, \end{matrix}$ now, using Hölder inequality on the right-hand side we obtain $\begin{matrix} {‖ λ^{ε} ‖}_{0, Γ^{ε}} ⩽ {‖ g^{ε} ‖}_{0, Γ^{ε}} . \end{matrix}$ Since $g^{ε} (x) = ε g (x / ε)$ , a.e. $x \in Γ^{ε}$ , then by periodicity and by summing up all the integrals on $Γ_{κ, ε}$ , $κ \in I$ , we have (after a change of variable and using that $card (I) = O (ε^{- 2})$ ) that $\begin{matrix} (31) & {‖ g^{ε} ‖}_{0, Γ^{ε}} = {(\sum_{κ \in I} \int_{Γ_{κ, ε}} {| ε g (\frac{x}{ε}) |}^{2} d Γ)}^{1 / 2} ⩽ C ε^{1 / 2} ‖ g ‖_{0, Γ}, \end{matrix}$ and then (29) holds.

To obtain a priori estimates for $u^{ε}$ we use $v^{ε} = u^{ε}$ as test function in (6), then we have $\begin{matrix} a^{ε} (u^{ε}, u^{ε}) + b^{ε} (λ^{ε}, u^{ε}) = ℓ^{ε} (u^{ε}) . \end{matrix}$ Now, using $μ^{ε} = 0$ in (7) we have that $b^{ε} (λ^{ε}, u^{ε}) ⩾ 0$ , then $\begin{matrix} a^{ε} (u^{ε}, u^{ε}) ⩽ ℓ^{ε} (u^{ε}) . \end{matrix}$ Using Poincaré inequality (notice that the constant does not depend on ε since we consider functions that vanish only on $\partial Ω$ ), and the ellipticity of $a^{ε}$ , it follows that $\begin{matrix} (32) & {| u^{ε} |}_{1, Ω^{ε}} ⩽ C ‖ f ‖_{0, Ω} . \end{matrix}$ From (32), and using again the Poincaré inequality, we obtain the a priori estimate (28). □

Note that, if we denote by $\tilde{u^{ε}}$ (resp. $\tilde{\nabla u^{ε}}$ ) the extension by 0 (inside the holes) of $u^{ε}$ (resp. $\nabla u^{ε}$ ), then from (32), we have that ${\tilde{u^{ε}}}_{ε > 0}$ and ${\tilde{\nabla u^{ε}}}_{ε > 0}$ are also bounded sequences.
4.2.2. Passing to the two-scale limit

The aim of this section is to pass to the limit $ε \to 0$ directly in the mixed formulation (6)–(7), using two-scale convergence (see [1]) and a two-scale convergence result on periodic surfaces (see [3]). The late result states that if a sequence $μ^{ε}$ in $L^{2} (Γ^{ε})$ is such that $\begin{matrix} ε \int_{Γ^{ε}} {| μ^{ε} (x) |}^{2} d Γ ⩽ C, \end{matrix}$ with C a positive constant, independent of ε, then there exists $M^{0} \in L^{2} (Ω; L^{2} (Γ))$ such that, up to a subsequence, $μ^{ε}$ two-scale converges to $M^{0}$ in the sense that $\begin{matrix} (33) & lim_{ε \to 0} ε \int_{Γ^{ε}} μ^{ε} (x) ϕ (x, \frac{x}{ε}) d Γ = \int_{Ω} \int_{Γ} M^{0} (x, y) ϕ (x, y) d_{y} Γ d x \end{matrix}$ for any $ϕ \in C (\overline{Ω}; C_{#} (Y))$ .

Proof of Theorem 3.2.
Using the a priori estimates of Proposition 4.1 we have that $\tilde{λ^{ε}}$ defined, for all $ε > 0$ , by $\begin{matrix} \tilde{λ^{ε}} = ε^{- 1} λ^{ε}, \end{matrix}$ is such that $\tilde{λ^{ε}}$ belongs to $L^{2} (Γ^{ε})$ and, using (29), such that $\begin{matrix} ε {‖ \tilde{λ^{ε}} ‖}_{0, Γ^{ε}}^{2} ⩽ C ‖ g ‖_{0, Γ}^{2} . \end{matrix}$ That is, the hypotheses of Theorem 2.1 in [3] are satisfied. Therefore, there exists $L^{0} \in L^{2} (Ω; L^{2} (Γ))$ such that $\tilde{λ^{ε}}$ two-scale converges in the sense of (33), up to a subsequence, to $L^{0}$ .

We recall that from the a priori estimates for $u^{ε}$ of Proposition 4.1 we have that $\tilde{u^{ε}}$ and $\tilde{\nabla u^{ε}}$ are bounded by uniform constants. Hence, using Proposition 1.14(i) in [1], there exists $(u^{0}, u^{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (Y^{*}) / R)$ such that, up to a subsequence, $\begin{matrix} \tilde{u^{ε}} \overset{2}{⇀} χ (y) u^{0} (x) and \tilde{\nabla u^{ε}} \overset{2}{⇀} χ (y) (\nabla u^{0} + \nabla_{y} u^{1} (x, y)) . \end{matrix}$

Now we will prove that the two-scale limits are solution of the mixed two-scale problem (8)–(9). To do so, we use as test function $v^{ε}$ in (6) an admissible two-scale test function of the form $\begin{matrix} v^{ε} (x) = ϕ (x) + ε ϕ_{1} (x, x / ε), \end{matrix}$ with $(ϕ, ϕ_{1}) \in D (Ω) \times D (Ω; C_{#}^{\infty} (Y))$ . Using the two-scale convergence of $\tilde{\nabla u^{ε}}$ , we have $\begin{matrix} a^{ε} (u^{ε}, v^{ε}) \to \int_{Ω} \int_{Y} χ (y) (\nabla u^{0} (x) + \nabla_{y} u^{1} (x, y)) \cdot (\nabla ϕ (x) + \nabla_{y} ϕ_{1} (x, y)) d y d x . \end{matrix}$ We also have $\begin{matrix} ℓ^{ε} (v^{ε}) \to \int_{Ω} \int_{Y} χ (y) f (x) ϕ (x) d y d x . \end{matrix}$ Next, since $\begin{matrix} b^{ε} (λ^{ε}, v^{ε}) = ε \int_{Γ^{ε}} (ε^{- 1} λ^{ε}) v^{ε} d Γ, \end{matrix}$ then, using the two-scale convergence of $\tilde{λ^{ε}}$ , we have $\begin{matrix} b^{ε} (λ^{ε}, v^{ε}) \to \int_{Ω} \int_{Γ} L^{0} (x, y) ϕ (x) d_{y} Γ d x . \end{matrix}$ Therefore, by a density argument, $((u^{0}, u^{1}), L^{0})$ satisfies the variational Eq. (8) of the two-scale mixed problem introduced in Section 3.3.

To pass to the limit in (9) we use that $ξ^{ε} = u^{0} (x) + ε u^{1} (x, x / ε)$ is an admissible test function. In fact, using $v^{0} = 0$ as a test function in (8) we see that $u^{1}$ is a weak solution of the local problem (19)–(21) and then we can apply standard regularity results. Let $r^{ε} = u^{ε} - ξ^{ε}$ . Therefore (9) becomes $\begin{matrix} (34) & b^{ε} (μ^{ε} - λ^{ε}, ξ^{ε} + r^{ε}) ⩽ 0 . \end{matrix}$ Moreover if $μ^{ε} \in Q^{ε}$ then $\begin{matrix} {‖ μ^{ε} ‖}_{0, Γ^{ε}} ⩽ C ε^{1 / 2} ‖ g ‖_{0, Γ}, \end{matrix}$ as in the proof of Proposition 4.1, hence, there exists $M^{0} \in L^{2} (Ω; L^{2} (Γ))$ such that $\tilde{μ^{ε}} = ε^{- 1} μ^{ε}$ two-scale converges to $M^{0}$ (in the sense of (33)).

Next, developing (34), we have that $\begin{array}{rcl} 0 & ⩾ & b^{ε} (μ^{ε} - λ^{ε}, ξ^{ε} + r^{ε}) = ε \int_{Γ^{ε}} (\frac{μ^{ε}}{ε} - \frac{λ^{ε}}{ε}) (ξ^{ε} + r^{ε}) |_{Γ^{ε}} d Γ \\ = & ε \int_{Γ^{ε}} (\tilde{μ^{ε}} - \tilde{λ^{ε}}) ξ^{ε} |_{Γ^{ε}} d Γ + ε \int_{Γ^{ε}} (\tilde{μ^{ε}} - \tilde{λ^{ε}}) r^{ε} |_{Γ^{ε}} d Γ . \end{array}$ The first integral in the right-hand side passes to the limit using the two-scale convergences of $\tilde{μ^{ε}}$ and $\tilde{λ^{ε}}$ . Using Hölder inequality in the second integral in the right-hand side, that $r^{ε}$ goes to 0 strongly in $H^{1}$ (by adapting Theorem 2.6 in [1] and using (28)) and that $ε ‖ \tilde{μ^{ε}} - \tilde{λ^{ε}} ‖_{0, Γ^{ε}}^{2}$ is bounded, we see that this term converges to 0 as ε goes to 0. Hence, in the limit we obtain $\begin{matrix} (35) & \int_{Ω} \int_{Γ} (M^{0} (x, y) - L^{0} (x, y)) u^{0} (x) d_{y} Γ d x ⩽ 0 . \end{matrix}$ Moreover $M^{0}$ and $L^{0}$ belong to $Q$ (see below). Now, let $M^{0} \in C (\overline{Ω}; C_{#} (Y))$ such that $M^{0} \in Q$ and we will use $μ^{ε} (x) = ε M^{0} (x, x / ε)$ , $x \in Γ^{ε}$ , as a test function in (9). In fact, the regularity of $M^{0}$ implies that $μ^{ε} \in L^{2} (Γ^{ε})$ and, since $M^{0}$ belongs to the convex $Q$ and $M^{0} (x, x / ε) \overset{2}{⇀} M^{0} (x, y)$ (in the sense of (33)), then there exists $ε^{0} > 0$ such that $μ^{ε} \in Q^{ε}$ for all $ε < ε^{0}$ . Hence, we can pass to the limit as before and, by a density argument (recall that $Q$ is a closed convex and $C (\overline{Ω}; C_{#} (Y))$ is dense in $L^{2} (Ω; L^{2} (Γ))$ ), we conclude that (35) holds for all $M^{0} \in Q$ .

Finally, we have to verify that the two-scale limit $L^{0}$ belongs to $Q$ . In fact, since $λ^{ε} \in Q^{ε}$ we have that $λ^{ε}$ verifies (30). Let $ψ \in C (\overline{Ω}; C_{#} (Γ))$ , then $φ^{ε}$ defined by $φ^{ε} (x) = ψ (x, x / ε)$ belongs to $L^{2} (Γ^{ε})$ , therefore $\begin{matrix} \int_{Γ^{ε}} λ^{ε} (x) ψ (x, \frac{x}{ε}) d Γ + \int_{Γ^{ε}} g^{ε} (x) | ψ (x, \frac{x}{ε}) | d Γ ⩾ 0 . \end{matrix}$ Using the two-scale convergence of $ε^{- 1} λ^{ε}$ to $L^{0}$ we can pass to the limit in the first term of the left-hand side. For the second term, we use (31), then $\tilde{g^{ε}} (x) = g (x / ε)$ is such that $ε ‖ \tilde{g^{ε}} ‖_{0, Γ^{ε}}^{2} ⩽ C$ and, since $g \in C_{#} (Y)$ , then it two-scale converges to $g^{0} (x, y) = g (y)$ . Hence, in the limit, we obtain $\begin{matrix} \int_{Ω} \int_{Γ} L^{0} (x, y) ψ (x, y) d x d_{y} Γ + \int_{Ω} \int_{Γ} g (y) | ψ (x, y) | d_{y} Γ d x ⩾ 0 . \end{matrix}$ By density, this inequality holds for any $ψ \in L^{2} (Ω; L^{2} (Γ))$ . In particular we have $\begin{matrix} \int_{Ω} \int_{Γ} L^{0} (x, y) ψ (x) d x d_{y} Γ + \int_{Ω} \int_{Γ} g (y) | ψ (x) | d_{y} Γ d x ⩾ 0 \end{matrix}$ for all $ψ \in L^{2} (Ω)$ , that is, $L^{0} \in Q$ . This completes the proof of the theorem. □

4.3. Well-posedness of the mixed two-scale problem

We show here that the two-scale mixed variational problem has a unique solution. Toward this end, and recalling Remark 3.1, let $R = L^{2} (Ω; L_{0}^{2} (Γ))$ and we consider the quotient space $L^{2} (Ω; L^{2} (Γ)) / R$ , that is, the space of equivalence classes in which all the functions in one class have the same mean value on Γ. We have that there exists an isomorphism between $L^{2} (Ω; L^{2} (Γ)) / R$ and $L^{2} (Ω)$ . In fact, let $T$ be defined by $\begin{array}{rcl} T : L^{2} (Ω; L^{2} (Γ)) / R \to L^{2} (Ω), \\ [M] \mapsto \int_{Γ} M (\cdot, y) d_{y} Γ, \end{array}$ where $[M]$ denotes the equivalence class of M. We easily see that $T$ is a bijection with an inverse defined by $T^{- 1} (M) = [| Γ |^{- 1} M] \in L^{2} (Ω; L^{2} (Γ)) / R$ , for all $M \in L^{2} (Ω)$ . Moreover, $T$ and $T^{- 1}$ are continuous maps. Therefore instead of looking for a Lagrange multiplier $L^{0}$ in $Q \subset L^{2} (Ω; L^{2} (Γ)) / R$ we will seek for $L^{0}$ in the convex subset $K$ of $L^{2} (Ω)$ defined by $\begin{matrix} (36) & K = {M \in L^{2} (Ω) : | Γ |^{- 1} M \in Q} \end{matrix}$ ( $| Γ |^{- 1} M$ is the class representative of $T^{- 1} (M)$ ). Note that, by the definition of $Q$ , the construction of the convex set $K$ is independent of the choice of the class representative.

Let ${\tilde{b}}_{h} : L^{2} (Ω) \times V \to R$ be the bilinear form defined by $\begin{matrix} {\tilde{b}}_{h} (M^{0}, (v^{0}, v^{1})) = \int_{Ω} M^{0} (x) v^{0} (x) d x \end{matrix}$ for all $M^{0} \in L^{2} (Ω)$ and all $(v^{0}, v^{1}) \in V$ . Hence, the equivalent homogenized two-scale mixed variational problem reads now:

Find $((u^{0}, u^{1}), L^{0}) \in V \times K$ such that $\begin{array}{rcl} (37) & a_{h} ((u^{0}, u^{1}), (v^{0}, v^{1})) + {\tilde{b}}_{h} (L^{0}, (v^{0}, v^{1})) = ℓ_{h} ((v^{0}, v^{1})), \\ (38) & {\tilde{b}}_{h} (M^{0} - L^{0}, (u^{0}, u^{1})) ⩽ 0 \end{array}$ for all $(v^{0}, v^{1}) \in V$ and all $M^{0} \in K$ .

The following inf–sup condition holds:

Proposition 4.2.
There exists $β > 0$ such that $\begin{matrix} inf_{L \in L^{2} (Ω)} sup_{(v, v^{1}) \in V} \frac{{\tilde{b}}_{h} (L, (v, v^{1}))}{‖ L ‖_{- 1, Ω} ‖ (v, v^{1}) ‖_{1}} ⩾ β . \end{matrix}$
Proof.
Let $L \in L^{2} (Ω) \subset H^{- 1} (Ω)$ . Using the Riesz representation theorem, there exists $\hat{v} \in H_{0}^{1} (Ω)$ such that $\begin{matrix} \int_{Ω} L (x) v (x) d x = {⟨ L, v ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)} = (\hat{v}, v) \end{matrix}$ (where $(\cdot, \cdot)$ denotes the inner product in $H_{0}^{1} (Ω)$ ), and $‖ \hat{v} ‖_{1, Ω} = ‖ L ‖_{- 1, Ω}$ . Let ${\hat{v}}^{1} = 0$ , then $‖ (\hat{v}, {\hat{v}}^{1}) ‖_{1} ⩽ C ‖ L ‖_{- 1, Ω}$ with $C = 1$ . Hence $\begin{matrix} {\tilde{b}}_{h} (L, (v, v^{1})) = \int_{Ω} L (x) \hat{v} (x) d x = (\hat{v}, \hat{v}) = ‖ \hat{v} ‖_{1, Ω}^{2} = ‖ L ‖_{- 1, Ω}^{2} ⩾ \frac{1}{C} ‖ L ‖_{- 1, Ω} {‖ (\hat{v}, {\hat{v}}^{1}) ‖}_{1} . \end{matrix}$ Therefore the inf–sup condition holds with $β = 1 / C = 1$ . □

Since Proposition 4.2 holds and the bilinear form $a_{h}$ is elliptic (see [1]) then, following [10], the equivalent homogenized mixed variational problem (37)–(38) has a unique solution. Remark 4.2.
If we replace $Q$ by the following convex set $\begin{array}{rcl} \tilde{Q} & = & {M^{0} \in L^{2} (Ω; L^{2} (Γ)) : \int_{Ω} \int_{Γ} M^{0} (x, y) ψ (x, y) d_{y} Γ d x \\ + \int_{Ω} \int_{Γ} g (y) | ψ (x, y) | d_{y} Γ d x ⩾ 0, \forall ψ \in L^{2} (Ω; L^{2} (Γ))}, \end{array}$ then we could have uniqueness of the Lagrange multiplier $L^{0}$ (instead of uniqueness up to an additive function $c (x, y)$ with 0 mean on Γ), since if $L^{0} \in \tilde{Q}$ implies that $L^{0} + c \in \tilde{Q}$ if and only if $\begin{matrix} \int_{Ω} \int_{Γ} c (x, y) ψ (x, y) d_{y} Γ d x = 0 \forall ψ \in L^{2} (Ω; L^{2} (Γ)), \end{matrix}$ that is $c (x, y) = 0$ . However, the mixed variational problem in this case would be ill-posed since the inf–sup condition for the bilinear form $b_{h}$ using the spaces $L^{2} (Ω; L^{2} (Γ))$ and $H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (Y^{}))$ does not hold.1
¹
If $L \in L^{2} (Ω; L^{2} (Γ))$ is such that $\int_{Γ} L (\cdot, y) d_{y} Γ = 0$ then $b_{h} (L, (v^{0}, v^{1})) = 0$ for all $(v^{0}, v^{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (Y^{}))$ .

Remark 4.3.
It is possible to pass to the limit as $ε \to 0$ in the variational inequality (4) using the two-scale convergence of $u^{ε}$ and the convergence results for convex functionals in [16,17], and prove that $u^{0}$ will be solution of the homogenized variational inequality (27). In this simple problem, we note that from (27), and following [10], we can derive directly a mixed homogenized variational formulation (see (41)–(42) in next section), suitable for numerical calculations. However, in a forthcoming paper in which we will study the homogenization of the Stokes problem in a periodically perforated domain with friction type boundary conditions (see [8]) we will note that the derivation of a homogenized mixed variational formulation from the homogenized variational inequality will not be as straightforward as in the present case.

5. Numerical results using the mixed variational formulations

In this section, we present some numerical tests in order to compare the heterogeneous solution of the mixed variational problem (6)–(7) with the homogenized solution of the two-scale problem (37)–(38). To do so, from (37)–(38) we derive a macroscopic problem posed in Ω and a periodic local problem posed in the perforated cell $Y^{*}$ . The macroscopic problem is obtained using $ϕ_{1} = 0$ as a test function in (8)–(9), in fact we have $\begin{array}{rcl} \int_{Ω} (\nabla u^{0} (x) + \int_{Y^{*}} \nabla_{y} u^{1} (x, y) d y) \cdot \nabla ϕ (x) d x \\ (39) & + \int_{Ω} L^{0} (x) ϕ (x) d x = \int_{Ω} \int_{Y^{*}} f (x) ϕ (x) d y d x, \\ (40) & \int_{Ω} (M^{0} (x) - L^{0} (x)) u^{0} (x) d x ⩽ 0 \end{array}$ for all $ϕ \in H_{0}^{1} (Ω)$ and all $M^{0} \in K \subset L^{2} (Ω)$ . The local problem is obtained using $v^{0} = 0$ in (37). In fact, after integration by parts, we recover problem (16)–(18). As usual, using a superposition argument, the solution $u^{1}$ of (16)–(18) can be written in terms of $u^{0}$ and the solutions of cell problems $w_{i}$ as in, for example, [1]. Doing so, we can rewrite the homogenized problem (39)–(40) for $u^{0}$ as $\begin{array}{rcl} (41) & \int_{Ω} (A_{h} \nabla u^{0} (x)) \cdot \nabla ϕ (x) d x + \frac{1}{| Y^{*} |} \int_{Ω} L^{0} (x) ϕ (x) d x = \int_{Ω} f (x) ϕ (x) d x, \\ (42) & \int_{Ω} (M^{0} (x) - L^{0} (x)) u^{0} (x) d x ⩽ 0, \end{array}$ where $A_{h}$ is the homogenized coefficients’ matrix. It is easy to see that this mixed variational inequality is well-posed since an inf–sup condition and an ellipticity condition hold.

Let us study the convex set $K$ defined in (36). From the definition of $Q$ we have that $M \in K$ if and only if $\int_{Ω} \int_{Γ} | Γ |^{- 1} M (x) ψ (x) d_{y} Γ d x + \int_{Ω} \int_{Γ} g (y) | ψ (x) | d_{y} Γ d x ⩾ 0$ , for all $ψ \in L^{2} (Ω)$ , therefore, in a strong form, we have $\begin{matrix} K = {M \in L^{2} (Ω) : | M (x) | ⩽ \int_{Γ} g (y) d_{y} Γ a.e. x \in Ω} . \end{matrix}$

For the approximation of both heterogeneous and homogenized problems and the periodic local problems (in order to calculate the homogenized coefficients), we use the Finite Element Method. For example, the discretization of the homogenized macroscopic problem (41)–(42), using the finite element space of continuous piece-wise linear functions, leads to the following matrix problem:

Find $u \in R^{N}$ and $l \in K$ such that $\begin{array}{rcl} A u + B l = f, \\ u^{t} B (m - l) ⩽ 0 \forall m \in K, \end{array}$ where N is the dimension of the finite element space and $K = {m \in R^{N} : | m_{i} | ⩽ \frac{1}{| Y^{*} |} \int_{Γ} g (y) d_{y} Γ, i = 1, \dots, N}$ (that is, a point-wise discretization of the convex $K$ ). Since A is invertible we can eliminate u and we obtain $\begin{matrix} l^{t} B^{t} A^{- 1} B (m - l) ⩾ f^{t} A^{- 1} B (m - l), \end{matrix}$ which corresponds to the optimality condition of the following constrained quadratic programming problem: $\begin{matrix} min_{m \in K} \frac{1}{2} m^{t} (B^{t} A^{- 1} B) m - f^{t} (A^{- 1} B) m . \end{matrix}$ For the resolution of this problem we use an implementation of the algorithm in [13]. For the construction of the finite element spaces and matrices we use the Finite Element Software FreeFEM++ (see [12]). The approximation of the heterogeneous mixed problem leads to a similar minimization problem but it requires the use of a finite element space defined on the periodic boundary $Γ^{ε}$ .

5.1. Numerical results

For the numerical tests the macroscopic domain is $Ω = {] 0, 1 [}^{2}$ and the perforation T is a disc of radius $r = 0.2$ . The right-hand side function f is defined by $\begin{matrix} f (x_{1}, x_{2}) = 2 π^{2} (sin (π x_{1}) sin (π x_{2})) \forall x = (x_{1}, x_{2}) \in Ω . \end{matrix}$ The threshold is given by a constant function g defined on Γ. Three threshold values were used. Since the threshold g is constant, the homogenized threshold is given by the constant function in Ω $\begin{matrix} x \in Ω \mapsto \frac{1}{| Y^{*} |} \int_{Γ} g (y) d_{y} Γ = \frac{| Γ |}{| Y^{*} |} g . \end{matrix}$

In the first test, the number of perforations is fixed to $N_{p} = 20 \times 20$ . Therefore, $ε = 1 / 20$ and $g^{ε} = ε g = (1 / 20) g$ for all $x \in Ω$ .

Fig. 1.

Iso values of $u^{ε}$ for constant threshold function $g = 4$ (top-left), $g = 7$ (top-center), and $g = 10$ (top-right). Iso values of the homogenized solution $u^{0}$ for $g = 4$ (middle-left), $g = 7$ (middle-center), and $g = 10$ (middle-right). Iso values of the corrected homogenized solution $u^{0} + ε u^{1}$ for $g = 4$ (bottom-left), $g = 7$ (bottom-center), and $g = 10$ (bottom-right). (Colors are visible in the online version of the article; https://dx-doi-org.web.bisu.edu.cn/10.3233/ASY-151346.)

Figure 1 presents the iso values of the numerical solutions for three values of g. We can observe that the homogenized solution $u^{0}$ has the same qualitative behavior than the heterogeneous solution $u^{ε}$ . From Table 1, we note that the maximal values of the homogenized solution $u^{0}$ are close to those of the heterogeneous solution $u^{ε}$ .

Next, we perform a convergence study. For the three values of g, we vary the number of perforations $N_{p}$ from $5 \times 5$ to $40 \times 40$ . As expected, from Table 2, we see that the $L^{2}$ convergence order is close to $O (ε)$ . In Table 3 we present the error $| u^{ε} - (u^{0} + ε u^{1}) |_{1, Ω^{ε}}$ (that is, we compare $u^{ε}$ to the corrected homogenized solution in the $H^{1}$ semi-norm). Here the convergence order is close to $O (\sqrt{ε})$ in the case $g = 4$ (as expected due to the presence of boundary layers near $\partial Ω$ , see [2], as the solution varies from zero, on $\partial Ω$ to non-zero values on a thin layer, see Fig. 1, top-left) but it improves to order $O (ε)$ in the case $g = 10$ (that is, in the case where the threshold function is not reached in a big portion of Ω and then the solution is equal to zero in a thicker layer, see Fig. 1, bottom-right).

Table 1

Maxima of the solutions, case $N_{p} = 20 \times 20$

	$g = 4$	$g = 7$	$g = 10$
$u^{ε}$	0.648554	0.312221	0.0950534
$u^{0}$	0.648700	0.307533	0.0889144
$u^{0} + ε u^{1}$	0.644944	0.305749	0.0883985

Table 2

$L^{2}$ error $‖ u^{ε} - u^{0} ‖_{0, Ω^{ε}}$ and rate of convergence α (calculated as the slope of the linear regression straight line passing through the points in the log–log graphic)

ε	$g = 4$	$g = 7$	$g = 10$
0.2	1.291e−02	1.692e−02	2.454e−02
0.1	5.686e−03	5.725e−03	6.958e−03
0.05	2.742e−03	2.627e−03	2.328e−03
0.025	1.377e−03	1.747e−03	1.224e−03
α	1.074	1.095	1.455

Table 3

$H^{1}$ semi-norm error $| u^{ε} - (u^{0} + ε u^{1}) |_{1, Ω^{ε}}$ and rate of convergence α

ε	$g = 4$	$g = 7$	$g = 10$
0.2	3.408e−01	4.414e−01	4.907e−01
0.1	2.161e−01	2.362e−01	2.499e−01
0.05	1.696e−01	1.341e−01	1.268e−01
0.025	1.560e−01	9.090e−02	6.625e−02
α	0.373	0.766	0.965

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