# Asymptotic analysis of abstract two-scale wave propagation problems

Sébastien Imperiale

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Sébastien Imperiale. Asymptotic analysis of abstract two-scale wave propagation problems. 2024. hal-03306856v2

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https://inria.hal.science/hal-03306856v2

Preprint submitted on 3 May 2024

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# Asymptotic Analysis of a class of abstract stiff wave propagation problems

Sébastien Imperiale

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## To cite this version:

Sébastien Imperiale. Asymptotic Analysis of a class of abstract stiff wave propagation problems. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2025, 23 (3), pp.1389-1416. 10.1137/24M1679458. hal-03306856v4

HAL Id: hal-03306856

https://inria.hal.science/hal-03306856v4

Submitted on 12 Sep 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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# ASYMPTOTIC ANALYSIS OF ABSTRACT TWO-SCALE WAVE PROPAGATION PROBLEMS WITH APPLICATIONS *

S. IMPERIALE †

**Abstract.**

This work addresses the mathematical analysis, by means of asymptotic analysis, of linear wave propagation problems involving two scales, represented by a single small parameter, and written in an abstract setting. This abstract setting is defined using linear operators in Hilbert spaces. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem. Applications in elastodynamics and piezoelectricity are presented.

**Key word.** Wave equation, Asymptotic analysis, Operator theory

**AMS subject classifications.** 35L05, 35B40, 47A50

**1. Introduction.** In the analysis of life science problems or industrial problems by means of partial differential equations (PDE), a class of problem that is frequently encountered is the class of two-scales problems. These problems – after non-dimensionnalisation – involve a unitless small parameter – denoted $\delta$ in this work. This parameter represents a ratio of two length scales or of two representative speed of propagations for instance. In the most interesting cases, when $\delta$ is small, the behavior of the underlying PDE is not trivially understood and some kind of asymptotic analysis process is required. More precisely, one study the limit of the solutions to the PDEs when $\delta$ goes to zero to obtain at the limit simpler PDEs in which the small parameter $\delta$ is missing. In applied mathematics such problems are very well understood for linear problems, see for instance [3] in the context of periodic structures and homogenization, [8] for the study of the mechanical behavior of plates (i.e. thin domains) or [18, 10] for the study of Stokes equations in nearly incompressible fluids. Many generalizations of these works have been published since then, see for instances [7, 5, 1].

In the semi-group theory the definition of an abstract framework based on linear operator theory [15] is now very standard (see [2, 9] or [19, 4] in the context of control theory). However, very few works in the literature propose a similar abstract framework, based on linear operator theory, (see however [12, 23]) for the analysis of some of the previously mentioned problems. This is precisely the objective of this work. Of course the mentioned class of problems is too large to be meaningfully represented in a single abstract framework, therefore the proposed analysis is restricted to wave propagation problems that possess what is denoted later a two-scale structure. The abstract framework that is presented in this work encompasses many relevant applications in elastodynamics, electromagnetic or piezoelectric problems (see [13, 14, 6]). The focus of this paper is the description of the abstract framework and its rigorous analysis, and present its application to some problems in elastodynamics and piezoelectricity. Note that although the proposed framework can be used to deal with homogenization problems, it is not a specific focus of this work since many dedicated mathematical tools already exists (see [20, 21, 22] to cite a few) to treat this question.

<footnote>*Submitted to the editors May 3, 2024.</footnote>

<footnote>†sebastien.imperiale@inria.fr. Inria, France.</footnote>
# ASYMPTOTIC ANALYSIS OF A CLASS OF ABSTRACT STIFF WAVE PROPAGATION PROBLEMS *

S. IMPERIALE †

**Abstract.**

This work addresses the mathematical analysis, by means of asymptotic analysis, of a class of linear wave propagation problems with singular stiff terms represented by a single small parameter. This abstract setting is defined using linear operators in Hilbert spaces. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem. Applications in nearly incompressible elastodynamics, elastic waves in thin plates, piezoelectricity, and homogenization are presented.

**Key word.** Wave equation, Asymptotic analysis, Operator theory

**AMS subject classifications.** 35L05, 35B40, 47A50

## 1. Introduction.

1.1. **Context.** In the analysis of life science problems or industrial problems by means of partial differential equations (PDEs), a class of stiff problems that is frequently encountered is the class of two-scale problems. These problems, after nondimensionalization, involve a unitless small parameter – denoted $\delta$ in this work. This parameter may represent a ratio of two length scales or of two representative speeds of propagation. In the most interesting cases, where $\delta$ is small, the PDEs are stiff and thus challenging to solve numerically. However, the limit PDEs derived from asymptotic analysis are typically easier to solve. As a result, solving the limit problem may be more advantageous, making it crucial to understand how the limit problem relates to the stiff one.

The asymptotic analysis of such problems is very well understood for linear PDEs problems; see for instance [3] in the context of periodic structures and homogenization; [8] for the study of the mechanical behavior of plates (i.e., thin domains) or [18, 10] for the study of Stokes equations in nearly incompressible fluids. Many generalizations of these works have been published since then, see for instance [7, 5, 1].

In the semi-group theory the definition of an abstract framework based on linear operator theory [15] is now very standard (see [2, 9] or [19, 4] in the context of control theory). Also, many works in the homogenization literature – starting from [29] – have used operator theory to obtain convergence results and operator estimates, see for instance [31, 32] or [23, 27]. However, few works in the literature propose a similar abstract framework, based on linear operator theory, for the asymptotic analysis of problems with similar mathematical structure (see [12] for instance). Therefore, the objective of this work is to propose and analyse an abstract framework for linear wave propagation problems – in Laplace domain – that possess what is later denoted a two-scale structure. The abstract framework presented in this work encompasses many relevant applications in elastodynamics, electromagnetic, or piezoelectric problems (see [13, 14, 6]). The focus of this paper is the description of the abstract framework and its rigorous analysis, and to present its application to some problems in elastodynamics and piezoelectricity. Note that the proposed framework can be used to deal

<footnote>*Submitted to the editors June 25, 2025.</footnote>

<footnote>†sebastien.imperiale@inria.fr. Inria, France.</footnote>
This article is organized as follows:

* Section 2 is devoted to the abstract definition of two-scale wave propagation problems and their asymptotic limits.

* Section 2.1 is dedicated to the definition of sequences of two-scale operators in Hilbert spaces. Formally these operators have the following form

$$G_{\delta} = \overline{\delta^{-1} G_{0} + G_{1}},$$

where $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators in Hilbert spaces. In this section we also introduce the notion of weak scale separation property. This definition corresponds to assumptions on $G_0$ and $G_1$ that enable an analysis of the limit process when $\delta$ goes to zero.

* In Section 2.2 a family of wave propagation problems is introduced. In particular, a simplified formal definition of these problems is

$$s^2 u_{\delta} + G_{\delta}^* G_{\delta} u_{\delta} = f_{\delta},$$

where $s = i\omega + \eta$ with $\eta > 0$ and $\omega \in \mathbb{R}$. Weak and strong convergence results towards the limit problems are provided. The limit problem reads, formally,

$$s^2 u_0 + \overline{Q_0 G_1}^* \overline{Q_0 G_1} u_0 = f_0,$$

where $Q_0$ is the orthogonal projection operator on the kernel of $G_0^*$. We also introduce in this section the (self-explanatory) notion of propagating and non-propagating solutions, and discussed as well some extensions to a more general class of problems. We conclude by introducing the notion of strong and strict scale separations used in Section to prove convergence estimates.

* Section 3 is devoted to the application of the proposed framework to three problems in physics.

* Section 3.1 tackles the problem of wave propagation in tissues, in particular, the limit from compressible to incompressible elastodynamics is discussed.

* Section 3.2 addresses the propagation of elastic waves in thin plates.

* Section 3.3 concerns the quasi-static approximation in piezo-electricity.

* Section 4 is devoted to establishing convergence estimate of the solutions of the family of problems towards the limit problems when the strong or strict scale separation property holds.

To ease the reading the some proofs of Section 2 are given Section 5.
with homogenization problems, but it is not a specific focus of this work since many dedicated mathematical tools already exists (see [20, 21, 22, 29, 30]) to treat this question. Still, an application example in 1D-periodic homogenization is presented.

## 1.2. Mathematical motivation.
In mathematical terms, the present work concerns the analysis of a family of wave propagation problems posed in the Laplace domain and parametrized by a small parameter $\delta \in (0, 1)$, which is intended to tend to zero. The problems are formally given by the equation

$$ (1.1) \qquad s^2 u_{\delta} + G_{\delta}^* G_{\delta} u_{\delta} = f, $$

where $s = i\omega + \eta$ with $\eta > 0$ and $\omega \in \mathbb{R}$ is a parameter, $u_{\delta}$ is the sought solution, $f$ is a source term, $G_{\delta}$ is a linear operator in Hilbert spaces that has formally the specific structure

$$ G_{\delta} = \delta^{-1} G_{0} + G_{1}, $$

and finally, in (1.1), $G_{\delta}^*$ is the adjoint of $G_{\delta}$. Studying the limit behavior of $u_{\delta}$ with respect to $\delta$ is a classical topic in asymptotic analysis. It can be done for instance by using the following Ansatz on $u_{\delta}$,

$$ u_{\delta} = u_{0} + \delta u_{1} + \delta^{2} u_{2} + \dots $$

Injecting this Ansatz in (1.1) and identifying terms with the same powers of $\delta$ leads to a cascade of equations. The first three equations are given by

$$ (1.2) \qquad \left\{ \begin{array}{l} \delta^{-2} \rightarrow G_0^* G_0 u_0 = 0, \\ \delta^{-1} \rightarrow G_0^* G_0 u_1 + (G_0^* G_1 + G_1^* G_0) u_0 = 0, \\ \delta^0 \rightarrow s^2 u_0 + G_1^* G_1 u_0 + (G_0^* G_1 + G_1^* G_0) u_1 + G_0^* G_0 u_2 = f. \end{array} \right. $$

Manipulating the first two equations leads to the following results

$$ (1.3) \qquad u_0 \in \mathrm{Ker} G_0, \quad u_1 = -(G_0^* G_0)^+ G_0^* G_1 u_0 + u_{1,0}, $$

where $(G_0^* G_0)^+$ is the (formal) Moonrose pseudo-inverse of $G_0^* G_0$ and $u_{1,0}$ is an undetermined function in $\mathrm{Ker} G_0$. Now, in order to get rid of the term $u_2$ in the last equation of (1.2) it is tested with a function $v \in \mathrm{Ker} G_0$,

$$ (1.4) \qquad s^2(u_0, v)_{\mathcal{H}} + (G_1 u_0, G_1 v)_{\mathcal{G}} + (G_0 u_1, G_1 v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}, $$

where $\mathcal{H}$ and $\mathcal{G}$ are some Hilbert spaces. In (1.4), $u_1$ can be substituted using (1.3),

$$ (1.5) \qquad s^2(u_0, v)_{\mathcal{H}} + (Q_0 G_1 u_0, G_1 v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}, \quad Q_0 = I - G_0 (G_0^* G_0)^+ G_0^* $$

It turns out – by the properties of the pseudo inverse – that $Q_0$ is the orthogonal projector into $\mathrm{Ker} G_0^*$. Hence $Q_0^2 = Q_0$ and $Q_0^* = Q_0$ and the equation (1.5) on $u_0$ can be written under the following symmetric form

$$ (1.6) \qquad s^2(u_0, v)_{\mathcal{H}} + (Q_0 G_1 u_0, Q_0 G_1 v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}. $$

The main objective of this work is to give general assumptions on functional spaces and on the (unbounded) linear operators $G_0$ and $G_1$ such that one can show a convergence result from solutions to (1.1) to the solution of a well-defined limit problem similar to (1.6). Some extensions and applications are also presented and in some specific cases convergence estimates are provided.
## 2. Two-scale wave propagation problems.

**2.1. Preliminary definition: Two-scale operators.** We assume given a Hilbert space with values in $\mathbb{C}$,

$$ (\mathcal{H}; (\cdot, \cdot)_{\mathcal{H}}) \quad \text{and} \quad (\mathcal{G}; (\cdot, \cdot)_{\mathcal{G}}). $$

In the rest of this work $\delta$, is a real parameter and consider that for each $\delta \in (0, 1)$, a given densely defined, closed, unbounded operators. We introduce below the definition of a sequence of formal two-scale operators.

**DEFINITION 2.1.** Let $G_{\delta}: D(G_{\delta}) \subset \mathcal{H} \mapsto \mathcal{G}$ be a sequence of unbounded operators. It is said to be a sequence of formal two-scale operators when, for each $\delta \in (0, 1)$,

$$ (2.1) \qquad G_{\delta} = \delta^{-1} G_{0} + G_{1}, $$

where the operators $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators from $\mathcal{H}$ to $\mathcal{G}$.

The adjective “formal” is used here to mean that, apart from the apparent structure, no interesting property can be deduced for such sequence of operators. In fact, the domain of definition of $G_{\delta}$ is given by

$$ D(G_{\delta}) = D(G_{0}) \cap D(G_{1}) $$

and may be restricted to $\{0\}$. We therefore require a more useful definition of two-scale sequence of operators. For the rest of the work, $*$ is used to represent the adjoint – in Hilbert spaces – of densely defined operators.

**DEFINITION 2.2.** Let $G_{\delta}: D(G_{\delta}) \subset \mathcal{H} \mapsto \mathcal{G}$ be a sequence of unbounded operators. It is said to be a sequence of two-scale operators when, for each $\delta \in (0, 1)$,

$$ (2.2) \qquad G_{\delta} = \overline{\delta^{-1} G_{0} + G_{1}}, $$

where the operators $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators from $\mathcal{H}$ to $\mathcal{G}$, and

$$ (2.3) \qquad \begin{cases} D(G_0) \cap D(G_1) \text{ dense in } (D(G_0); \|\cdot\|_{\mathcal{H}} + \|G_0\|_{\mathcal{G}}), & (a) \\ D(G_0^*) \cap D(G_1^*) \text{ dense in } (D(G_0^*); \|\cdot\|_{\mathcal{G}} + \|G_0^*\|_{\mathcal{H}}), & (b) \end{cases} $$

In Definition 2.2 it is implicitly assumed that $\delta^{-1}G_0 + G_1$ is closable and that $G_{\delta}$ is the closure of this operator. Moreover, as a direct consequence of the fact that $G_0$ is densely defined and (2.3)(a), $G_{\delta}$ is densely defined.

**THEOREM 2.3.** For any sequence of two-scale operators $G_{\delta}$, let $\{u_{\delta}\}$ be a sequence in $D(G_{\delta})$, uniformly bounded in $\mathcal{H}$ and $\{g_{\delta}\}$ be a sequence in $D(G_{\delta}^*)$, uniformly bounded in $\mathcal{G}$, then

$$ (2.4) \qquad \begin{cases} \forall h \in \mathcal{D}(G_0^*) \cap \mathcal{D}(G_1^*), \quad (u_{\delta}, G_0^* h)_{\mathcal{G}} \underset{\delta \to 0}{\longrightarrow} 0 \quad \Rightarrow \quad u_{\delta} \underset{\mathcal{H}}{\longrightarrow} u_0 \in \text{Ker } G_0, & (a) \\ \forall v \in \mathcal{D}(G_0) \cap \mathcal{D}(G_1), \quad (g_{\delta}, G_0 v)_{\mathcal{G}} \underset{\delta \to 0}{\longrightarrow} 0 \quad \Rightarrow \quad g_{\delta} \underset{\mathcal{G}}{\longrightarrow} g_0 \in \text{Ker } G_0^*, & (b) \end{cases} $$

where the weak convergences hold up to subsequences.
**1.3. Main result.** As mentioned, the main result of this work is a strong convergence result of a family of solutions of parametrized stiff wave propagation problems. Below $\mathcal{H}$ and $\mathcal{G}$ are some Hilbert spaces and the operator $G^* : D(G^*) \subset \mathcal{G} \to \mathcal{H}$ denotes the adjoint of a densely defined operator $G : D(G) \subset \mathcal{H} \to \mathcal{G}$. Moreover, for a closable operator $G$, $\bar{G}$ denotes the closure of $G$. These notions are briefly recapped in Section 2.1.

**THEOREM 1.1.** For every $\delta \in (0,1)$, let $G_\delta : D(G_\delta) \subset \mathcal{H} \to \mathcal{G}$ be a linear operator such that

$$G_{\delta} = \overline{\delta^{-1} G_{0} + G_{1}},$$

where $G_0$ and $G_1$ are densely defined and closed linear operators from $\mathcal{H}$ to $\mathcal{G}$ with domain $D(G_0)$ and $D(G_1)$ respectively. Let $Q_0$ be the orthogonal projection operator on the closed subspace $\text{Ker } G_0^*$ of $\mathcal{G}$. When

$$ (1.7) \qquad \left\{ \begin{array}{ll} D(G_0) \cap D(G_1) \text{ dense in } D(G_0), & (a) \\ D(G_0^*) \cap D(G_1^*) \text{ dense in } D(G_0^*), & (b) \\ Q_0 G_1 \text{ is closable,} & (c) \\ \text{Ker } G_0 \cap D(G_1) \text{ dense in } \text{Ker } G_0 \cap D(\overline{Q_0 G_1}), & (d) \end{array} \right. $$

then, for any element $f \in \mathcal{H}$ and $s = i\omega + \eta$ with $\eta > 0$ and $\omega \in \mathbb{R}$, the sequence of $\{u_\delta\} \subset D(G_\delta)$ solutions to

$$\forall v \in D(G_{\delta}), \quad s^2(u_{\delta}, v)_{\mathcal{H}} + (G_{\delta}u_{\delta}, G_{\delta}v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}$$

satisfies

$$\|u_{\delta} - u_{0}\|_{\mathcal{H}} + \|\overline{Q_{0}G_{1}}(u_{\delta} - u_{0})\|_{\mathcal{G}} \xrightarrow{\delta \to 0} 0$$

where $u_0 \in \text{Ker } G_0 \cap D(\overline{Q_0 G_1})$ is the unique solution to

$$\forall v \in \text{Ker } G_0 \cap D(\overline{Q_0 G_1}), \quad s^2(u_0, v)_{\mathcal{H}} + (\overline{Q_0 G_1} u_0, \overline{Q_0 G_1} v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}.$$

**Remark 1.2.** In practice, as far as PDE problems are concerned, the density results (1.7)(a) and (1.7)(b) are easily verified using known density results of smooth functions that belongs typically to $D(G_0) \cap D(G_1)$ or $D(G_0^*) \cap D(G_1^*)$. Assumption (1.7)(c) seems natural since it allows to define a well posed limit problem in a space that is complete (see Section 2.1). Assumption (1.7)(d) is justified by the strategy used to prove convergence. In practice it is easily proved but seems not, in a general context, a consequence of the other three assumptions.

In the manuscript intermediate results are given with additional terms, or $\delta$-dependent source terms, however, Theorem 1.1 captures the essence of this work.
These two properties are proven very similarly. In Section 5 only the proof of (2.4)(a) is provided and it is a direct application of the density results (2.3). Note that Theorem 2.3 has a rather intuitive direct consequence: for a sequence $u_\delta$ converging towards $u_0$ in $\mathcal{H}$, when there exists $C > 0$, independent of $\delta$ such that

$$ (2.5) \qquad \|G_\delta u_\delta\|_G \le C \quad \text{then} \quad u_0 \in \text{Ker } G_0. $$

This property is still not sufficient for our purpose. In what follows three different properties, denoted respectively weak, strong and strict scale separation properties are introduced. These properties describe in more details the relation between the operators $G_0$ and $G_1$. In this section only the notion of weak scale separation is introduced.

**DEFINITION 2.4.** A sequence of two-scale operators $G_\delta$ satisfies the weak scale separation property if

$$ (2.6) \qquad \text{Ker } G_0^* \cap D(G_1^*) \text{ dense in } (\text{Ker } G_0^*; \|\cdot\|_G), $$

The fundamental consequence of assumption (2.6) is given below. To state the corresponding theorem the orthogonal projection operator in the kernel of $G_0^*$ is introduced. It is denoted $Q_0$ and is defined by,

$$ Q_0 \in \mathcal{L}(G), \quad Q_0^2 = Q_0, \quad (g - Q_0 g, h)_G = 0 \quad \forall (g, h) \in G \times \text{Ker } G_0^*. $$

Note that $Q_0$ is a self-adjoint operator.

**THEOREM 2.5.** For any sequence of two-scale operators $G_\delta$ satisfying the weak scale separation property then

$$ (2.7) \qquad G_1^* Q_0 \text{ is densely defined in } G \text{ and } Q_0 G_1 \text{ is closable} \\ \text{and} $$

$$ (2.8) \qquad Q_0 G_\delta \subset \overline{Q_0 G_1}. $$

Note that the fact that $Q_0 G_1$ is closable is essential in the analysis of problems involving the corresponding two-scale operator $G_\delta$. The reason being that this operator naturally appears in the forthcoming asymptotic analysis. The property of $Q_0 G_1$ of being closable is essential to obtain a well-posed limit problem in the complete Hilbert space $D(\overline{Q_0 G_1})$. Relation (2.8) means that $\overline{Q_0 G_1}$ is an extension of $Q_0 G_\delta$.

**2.2. A family of wave propagation problems and their asymptotic limits.** In this section we assume given a sequence of two-scale operators $G_\delta$ satisfying the weak scale separation property. To shorten the notation we define the Hilbert space,

$$ \mathcal{V}_\delta = D(G_\delta) \subset \mathcal{V}_0 = D(\overline{Q_0 G_1}), $$

equipped respectively with the graph norm of $G_\delta$ and $\overline{Q_0 G_1}$, and we point out that $\mathcal{V}_\delta \subset \mathcal{V}_0$ is a consequence of (2.8). We consider the following wave propagation problem in Laplace domain: for $f_\delta \in \mathcal{H}$ given, find $u_\delta \in \mathcal{V}_\delta$ such that, for all $v \in \mathcal{V}_\delta$,

$$ (2.9) \qquad s^2(u_\delta, v)_\mathcal{H} + (G_\delta u_\delta, G_\delta v)_G = (f_\delta, v)_\mathcal{H}, $$
**1.4. Organization of the article.** Section 2 is devoted to the abstract definition of a family of singular wave propagation problems and their asymptotic limits.

■ Section 2.1 gives a brief recap of the important notions in unbounded operators theory.

■ Section 2.2 is dedicated to the definition of sequences of two-scale operators in Hilbert spaces. These operators have the following form

$$G_{\delta} = \overline{\delta^{-1} G_{0} + G_{1}},$$

where $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators in Hilbert spaces.

■ In Section 2.3 the family of wave propagation problems is introduced. In particular, a simplified formal definition of these problems is

$$s^2 u_{\delta} + G_{\delta}^* G_{\delta} u_{\delta} = f_{\delta},$$

Weak and strong convergence results towards the limit problems are provided. The limit problem reads, formally,

$$s^2 u_0 + \overline{Q_0 G_1}^* \overline{Q_0 G_1} u_0 = f_0,$$

where $Q_0$ is the orthogonal projection operator on the kernel of $G_0^*$. We also introduce in this section the notion of propagating and non-propagating solutions, and discussed as well some extensions to a more general class of problems. We conclude by introducing the notion of strong and strict scale separations used in Section 5 to prove convergence estimates.

Section 3 concerns the study of a one-dimensional periodic homogenization problem. Although this problem is nowadays very well studied, convergence results in average – using the shift method of [30] – are obtained with very few requirements on the parameters regularity.

Section 4 is devoted to the application of the proposed framework to three problems in elastodynamics.

■ Section 4.1 tackles the problem of wave propagation in tissues, in particular, the limit from compressible to incompressible elastodynamics is discussed.

■ Section 4.2 addresses the propagation of elastic waves in thin plates.

■ Section 4.3 concerns the quasi-static approximation in piezoelectricity.

Section 5 is devoted to establishing convergence estimate of the solutions of the family of problems towards the limit problems when the strong or strict scale separation property holds.

To ease the reading, the proofs of some of the results from Section 2 and Section 3 are given Section 6.
where $s = i\omega + \eta$ with $\eta > 0$ and $\omega \in \mathbb{R}$. By application of the Lax-Milgram theorem one can show that this problem admits a unique solution in $V_\delta$. Moreover, choosing $v = s u_\delta$ as a test function in (2.9) one can show the following stability estimate

$$ (2.10) \qquad \|s u_\delta\|_{\mathcal{H}}^2 + \|G_\delta u_\delta\|_{\mathcal{G}}^2 \le \eta^{-2} \|f_\delta\|_{\mathcal{H}}^2. $$

Note that, such estimate can be used to derive estimates in time domain using the inverse Fourier transform. We refer the reader to [24] or [25] for more details.

### 2.2.1. Weak convergence towards the limit problem.

**THEOREM 2.6.** Assume that there exists $f_0 \in \mathcal{H}$ such that $f_\delta \underset{\mathcal{H}}{\to} f_0$, then, up to a subsequence,

$$ (2.11) \qquad u_\delta \underset{V_0}{\to} u_0, $$

where $u_0 \in \text{Ker } G_0 \cap V_0$, satisfies, for all $v \in \text{Ker } G_0 \cap D(G_1)$,

$$ (2.12) \qquad s^2(u_0, v)_{\mathcal{H}} + (\overline{Q_0 G_1} u_0, \overline{Q_0 G_1} v)_{\mathcal{G}} = (f_0, v)_{\mathcal{H}}. $$

When

$$ (2.13) \qquad \text{Ker } G_0 \cap D(G_1) \text{ is dense in } (\text{Ker } G_0 \cap V_0; \|\cdot\|_{V_0}), $$

then $u_0$ is uniquely defined by (2.12).

*Proof.* Since $f_\delta$ weakly converges in $\mathcal{H}$ it is uniformly bounded in $\mathcal{H}$ hence, from (2.10), there exists $C > 0$ independent of $\delta$ such that

$$ (2.14) \qquad \|u_\delta\|_{\mathcal{H}}^2 + \|G_\delta u_\delta\|_{\mathcal{G}}^2 \le C. $$

From (2.5) we deduce that there exists $u_0 \in \text{Ker } G_0$ such that, up to a subsequence,

$$ u_\delta \underset{\mathcal{H}}{\to} u_0. $$

For that subsequence, writing $I = (I - Q_0) + Q_0$ and with the orthogonality property of $Q_0$ we have

$$ \|u_\delta\|_{\mathcal{H}}^2 + \|\overline{Q_0 G_1} u_\delta\|_{\mathcal{G}}^2 \le \|u_\delta\|_{\mathcal{H}}^2 + \|Q_0 G_\delta u_\delta\|_{\mathcal{G}}^2 + \|(I - Q_0) G_\delta u_\delta\|_{\mathcal{G}}^2 \le C. $$

Hence, $u_\delta$ is uniformly bounded in $V_0$ and, up to a subsequence,

$$ (2.15) \qquad u_\delta \underset{V_0}{\to} u_0. $$

For that subsequence, we choose $v \in D(G_0) \cap D(G_1)$ in (2.9), we have, setting $g_\delta = G_\delta u_\delta$,

$$ (2.16) \qquad (g_\delta, G_0 v)_{\mathcal{G}} = \delta(f_\delta, v)_{\mathcal{H}} - \delta s^2(u_\delta, v)_{\mathcal{H}} - \delta(g_\delta, G_1 v)_{\mathcal{G}}. $$

Estimate (2.14) shows that $g_\delta$ is uniformly bounded in $\mathcal{G}$, hence, using (2.16) we can apply Theorem 2.3: there exists $g_0 \in \text{Ker } G_0^*$ such that, again, up to a subsequence,

$$ g_\delta \underset{\mathcal{G}}{\to} g_0. $$
## 2. A family of singular wave propagation problems.

### 2.1. Preliminaries on unbounded linear operators in Hilbert spaces.

The objective of this section is to recap briefly some classical results on unbounded operators. The reader could find more details on this topic in [15, 11] for instance. For the rest of the article $\mathcal{H}$ and $\mathcal{G}$ are Hilbert spaces with values in $\mathbb{C}$, with norms and scalar products denoted

$$ \|\cdot, \|_{\mathcal{H}}, (\cdot, \cdot)_{\mathcal{H}} \text{ and } \|\cdot, \|_{\mathcal{G}}, (\cdot, \cdot)_{\mathcal{G}}. $$

The notation $G : D(G) \subset \mathcal{H} \mapsto \mathcal{G}$ is used to represent a linear operator $G$ defined from a subspace (its domain) $D(G)$ of $\mathcal{H}$ to $\mathcal{G}$.

* The operator $G$ is densely defined if $D(G)$ is dense in $\mathcal{H}$ for the norm $\|\cdot\|_{\mathcal{H}}$.

* The operator $\tilde{G} : D(\tilde{G}) \subset \mathcal{H} \mapsto \mathcal{G}$ is an extension of $G$ if $D(G) \subset D(\tilde{G})$ and both operators coincide on $D(G)$. One writes $G \subset \tilde{G}$.

* The operator $G$ is closed if $D(G)$ is complete for the graph norm

$$ \| \cdot \|_{D(G)}^2 = \| \cdot \|_{\mathcal{H}}^2 + \| G \cdot \|_{\mathcal{G}}^2. $$

The essential notion of adjoint operators is recalled below.

* If $G$ is densely defined one can define the adjoint operator

$$ G^* : D(G^*) \subset \mathcal{G} \mapsto \mathcal{H}, \quad \forall (u, g) \in D(G) \times D(G^*) \quad (Gu, g)_{\mathcal{G}} = (u, G^*g)_{\mathcal{H}}. $$

The operator $G^*$ is closed and its kernel $\text{Ker } G^*$ is a closed subspace of $\mathcal{G}$.

* If $G$ is closed and densely defined then $G^*$ is also closed and densely defined and $\text{Ker } G$ is a closed subspace of $\mathcal{H}$.

Finally, closures of operators are used in this work. It corresponds to the minimal closed extension of a closable operator. More precisely:

* The operator $G$ is closable if it has a closed extension. Equivalently, one can verify that, for any sequence $\{u_n\} \subset D(G)$

$$ u_n \xrightarrow{\mathcal{H}} 0 \text{ and } Gu_n \xrightarrow{\mathcal{G}} g \Rightarrow g = 0. $$

* A closable operator has a minimal extension $\overline{G}$ called the closure. It is the closed operator such that every closed extension of $G$ is an extension of $\overline{G}$.

* If $G$ is densely defined and closable then $G^* = \overline{G}^*$, moreover $G^*$ is closed and densely defined and $G^{**} = \overline{G}$.

* If $G^*$ is densely defined then $G$ is closable.

Considering two linear operators $G_0 : D(G_0) \subset \mathcal{H} \mapsto \mathcal{G}$ and $G_1 : D(G_1) \subset \mathcal{H} \mapsto \mathcal{G}$ as well as a bounded operator $Q \in \mathcal{L}(\mathcal{G})$, then

* the domain of the operator $G_0 + G_1$ is $D(G_0) \cap D(G_1)$,

* the operator $QG$ satisfies $D(QG) = D(G)$ and $QG$ is densely defined when $G$ is, however $QG$ is not necessarily a closed operator even if $G$ is.
Finally, we choose $v \in \text{Ker } G_0 \cap \mathcal{D}(G_1)$ in (2.9), we have, passing to the limit

$$ (2.17) \quad s^2(u_\delta, v)_H + (G_\delta u_\delta, G_1 v)_G = (f_\delta, v)_H \Rightarrow s^2(u_0, v)_H + (g_0, G_1 v)_G = (f_0, v)_H. $$

It remains to characterize $g_0$. We have, thanks to (2.15),

$$ Q_0 g_\delta = Q_0 G_\delta u_\delta = \overline{Q_0 G_1} u_\delta \quad \Rightarrow \quad Q_0 g_0 = \overline{Q_0 G_1} u_0 $$

Since $Q_0 g_0 = g_0$, we have characterized $g_0$. We obtain (2.12) using (2.17) and by writing that, for all $v \in \text{Ker } G_0 \cap \mathcal{D}(G_1)$, we have

$$ (g_0, G_1 v)_G = (g_0, Q_0 G_1 v)_G = (g_0, \overline{Q_0 G_1} v)_G = (\overline{Q_0 G_1} u_0, \overline{Q_0 G_1} v)_G. $$

When (2.13) holds, by density, one can show that $u_0$ satisfies (2.12) for every test function in $\text{Ker } G_0 \cap \mathcal{V}_0$. One can conclude using the Lax-Milgram theorem. Note that, in that case, each subsequence converges towards the unique solution to problem (2.12). □

### 2.2.2. Quasi-static solutions and propagating solutions.
Assume for this section that (2.13) holds. We introduce the space $\mathcal{R}_0$ related to the kernel of $\overline{Q_0 G_1}$. More precisely, we define

$$ (2.18) \quad \mathcal{R}_0 := \text{Ker } \overline{Q_0 G_1} \cap \text{Ker } G_0 \cap \mathcal{V}_0 = \text{Ker } \overline{Q_0 G_1} \cap \text{Ker } G_0. $$

Since by definition $\overline{Q_0 G_1}$ is closed and densely defined in $\mathcal{H}$ its kernel is closed in $\mathcal{H}$. It is also closed in $\mathcal{V}_0$ since, for all $r \in \text{Ker } \overline{Q_0 G_1}$ we have $\|r\|_{\mathcal{V}_0} = \|r\|_{\mathcal{H}}$. We deduce then that the space $\mathcal{R}_0$ is a closed subspace of $\text{Ker } G_0 \cap \mathcal{V}_0$. We can therefore define $\mathcal{S}_0$ as the orthogonal complement of $\mathcal{R}_0$ in that space. We have

$$ (2.19) \quad \text{Ker } G_0 \cap \mathcal{V}_0 = \mathcal{R}_0 \oplus \mathcal{S}_0, $$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{V}_0$, more precisely, we have

$$ \forall (r, s) \in \mathcal{R}_0 \times \mathcal{S}_0, \quad 0 = (r, s)_{\mathcal{V}_0} = (r, s)_{\mathcal{H}} + (\overline{Q_0 G_1} r, \overline{Q_0 G_1} s)_G = (r, s)_{\mathcal{H}}, $$

hence $\mathcal{R}_0$ and $\mathcal{S}_0$ are also orthogonal with respect to the scalar product in $\mathcal{H}$. The space $\mathcal{R}_0$ is called the space of quasi-static solutions whereas the space $\mathcal{S}_0$ is called the space of propagating solutions. This terminology is justified below. For all $u_0$ solution to (2.12) we have

$$ u_0 = r_0 + s_0 $$

where $r_0 \in \mathcal{R}_0$ and $s_0 \in \mathcal{S}_0$ are solutions to

$$ \left\{ \begin{array}{ll} \forall r \in \mathcal{R}_0, & s^2(r_0, r)_{\mathcal{H}} = (f_0, r)_{\mathcal{H}}, \\ \forall v \in \mathcal{S}_0, & s^2(s_0, v)_{\mathcal{H}} + (\overline{Q_0 G_1} s_0, \overline{Q_0 G_1} v)_G = (f_0, v)_{\mathcal{H}}. \end{array} \right. $$

Up to a factor $s^2$, $r_0$ is the projection of $f_0$ on the space $\mathcal{R}_0$ whereas $s_0$ satisfies an abstract wave propagation problem similar to (2.12).
**2.2. Two-scale operators.** We first introduce below the definition of a sequence of formal two-scale operators.

**DEFINITION 2.1.** Let $G_\delta: D(G_\delta) \subset \mathcal{H} \mapsto \mathcal{G}$ be a sequence of unbounded operators. It is said to be a sequence of formal two-scale operators when, for each $\delta \in (0, 1)$,

$$ (2.1) \qquad G_{\delta} = \delta^{-1} G_{0} + G_{1}, $$

where the operators $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators from $\mathcal{H}$ to $\mathcal{G}$.

The adjective “formal” is used here to indicate that, apart from the apparent structure, no significant properties can be deduced for such a sequence of operators. In fact, the domain of definition of $G_\delta$ is given by

$$ D(G_{\delta}) = D(G_{0}) \cap D(G_{1}) $$

and may be reduced to $\{0\}$. Moreover, the operator $G_\delta$ may not be closed; hence, its domain may not be complete, which is essential for formulating the problem within a Hilbert space framework. Therefore, a more robust definition of sequences of two-scale operators is required.

**DEFINITION 2.2.** Let $G_\delta: D(G_\delta) \subset \mathcal{H} \mapsto \mathcal{G}$ be a sequence of unbounded operators. It is said to be a sequence of two-scale operators when, for each $\delta \in (0, 1)$,

$$ (2.2) \qquad G_{\delta} = \overline{\delta^{-1} G_{0} + G_{1}}, $$

where the operators $G_0$ and $G_1$ are $\delta$-independent, densely defined and closed operators from $\mathcal{H}$ to $\mathcal{G}$, and

$$ (2.3) \qquad \begin{cases} D(G_0) \cap D(G_1) \text{ dense in } (D(G_0); \|\cdot\|_{\mathcal{H}} + \|G_0\|_{\mathcal{G}}), & (a) \\ D(G_0^*) \cap D(G_1^*) \text{ dense in } (D(G_0^*); \|\cdot\|_{\mathcal{G}} + \|G_0^*\|_{\mathcal{H}}). & (b) \end{cases} $$

In Definition 2.2 it is implicitly assumed that $\delta^{-1}G_0 + G_1$ is closable and that $G_\delta$ is the closure of this operator. Moreover, as a direct consequence of the fact that $G_0$ is densely defined and (2.3)(a), $G_\delta$ is densely defined.

**THEOREM 2.3.** For any sequence of two-scale operators $G_\delta$, let $\{u_\delta\}$ be a uniformly bounded sequence in $\mathcal{H}$ and $\{g_\delta\}$ a uniformly bounded sequence in $\mathcal{G}$, then

$$ (2.4) \qquad \begin{cases} \forall h \in D(G_0^*) \cap D(G_1^*), (u_\delta, G_0^* h)_G \underset{\delta \to 0}{\longrightarrow} 0 \Rightarrow u_\delta \underset{\mathcal{H}}{\longrightarrow} u_0 \in \text{Ker } G_0, & (a) \\ \forall v \in D(G_0) \cap D(G_1), (g_\delta, G_0 v)_G \underset{\delta \to 0}{\longrightarrow} 0 \Rightarrow g_\delta \underset{\mathcal{G}}{\longrightarrow} g_0 \in \text{Ker } G_0^*, & (b) \end{cases} $$

where the weak convergences hold up to subsequences.

These two properties are proven very similarly. In Section 6 only the proof of (2.4)(a) is provided and it is a direct application of the density result (2.3)(b). Note that Theorem 2.3 has a rather intuitive direct consequence: for any sequence $\{u_\delta\} \subset D(G_\delta)$ converging towards $u_0$ in $\mathcal{H}$, then, if there exists $C > 0$, independent of $\delta$ such that

$$ (2.5) \qquad \|G_\delta u_\delta\|_{\mathcal{G}} \le C \quad \text{then} \quad u_0 \in \text{Ker } G_0. $$
### 2.2.3. Extensions: Lower order terms and more general source term.
several generalizations of the results given in the previous section can be studied. We focus here on the following wave propagation problem: for $\ell_{\delta} \in \mathcal{L}(V_0)$ given, find $u_{\delta} \in V_{\delta}$ such that, for all $v \in V_{\delta}$,

$$ (2.20) \qquad s^2(u_{\delta}, v)_{\mathcal{H}} + (G_{\delta} u_{\delta}, G_{\delta} v)_{\mathcal{G}} + sb(u_{\delta}, v) = \ell_{\delta}(v). $$

This problem involves, on the one hand, a sesquilinear form $b(\cdot, \cdot)$, that is continuous and positive in $V_0$: for all $u \in V_0$ and $v \in V_0$,

$$ (2.21) \qquad |b(u, v)| \le C_b \|u\|_{V_0} \|v\|_{V_0}, \quad \Re e(b(u, u)) \ge 0, $$

and, on the other hand, a sequence of anti-linear form $\ell_{\delta} \in \mathcal{L}(V_0)$ such that, for all,

$$ (2.22) \qquad \forall v \in V_0, \quad |\ell_{\delta}(v)| \le c_{\ell} \|v\|_{\mathcal{H}} + C_{\ell} \|\overline{Q_0 G_1} v\|_{\mathcal{G}}. $$

Solutions of (2.20) satisfy the following estimate (it is proven using $v = su$ as a test function),

$$ (2.23) \qquad \|s u_{\delta}\|_{\mathcal{H}}^2 + \|G_{\delta} u_{\delta}\|_{\mathcal{G}}^2 \le \frac{1}{\eta^2} (c_{\ell} + |s|C_{\ell})^2. $$

The sesquilinear form $b$ allows to take into account problems with dissipation, viscosity and/or couplings (see Section 3.3 on piezoelectricity for an application example). The anti-linear form $\ell_{\delta}$ is introduced to take into account problems with boundary data for instance. Both $\ell_{\delta}$ and $b$ may depend on $s$ (as well as the $c_{\ell}$, $C_{\ell}$ and $C_b$) in the case of equations obtained from the Laplace transform of time domain problems. We omit this dependence to simplify the reading. Theorem 2.6 can be extended to this case rather straightforwardly (the proof of the result below is therefore omitted).

**THEOREM 2.7.** Assume that there exists an anti-linear form $\ell_0 \in \mathcal{L}(V_0)$ such that

$$ \forall v \in \text{Ker } G_0 \cap D(G_1), \quad \ell_{\delta}(v) \underset{\delta \to 0}{\longrightarrow} \ell_0(v) $$

then, up to a subsequence,

$$ u_{\delta} \underset{V_0}{\longrightarrow} u_0. $$

where $u_0 \in \text{Ker } G_0 \cap V_0$, satisfies, for all $v \in \text{Ker } G_0 \cap D(G_1)$,

$$ (2.24) \qquad s^2(u_0, v)_{\mathcal{H}} + (\overline{Q_0 G_1} u_0, \overline{Q_0 G_1} v)_{\mathcal{G}} + sb(u_0, v) = \ell_0(v). $$

When $\text{Ker } G_0 \cap D(G_1)$ is dense in $(\text{Ker } G_0 \cap V_0; \| \cdot \|_{V_0})$, then $u_0$ is uniquely defined by (2.24).

### 2.2.4. Strong convergence.

**THEOREM 2.8.** Assume that there exists an anti-linear form $\ell_0 \in V_0$, such that

$$ (2.25) \qquad \|\ell_{\delta} - \ell_0\|_{\mathcal{L}(V_0)} \underset{\delta \to 0}{\longrightarrow} 0, $$

and assume that (2.13) holds, then, the sequence of solutions $\{u_{\delta}\} \subset V_0$ to (2.20), satisfies

$$ (2.26) \qquad u_{\delta} \underset{V_0}{\longrightarrow} u_0, $$

where $u_0$ is the unique solution to (2.24).
### 2.3. A family of wave propagation problems and their asymptotic limits.
In this section we assume given a sequence of two-scale operators $G_\delta$. To shorten the notation we define the Hilbert space,

$$V_\delta = D(G_\delta)$$

equipped with the graph norm of $G_\delta$ and the corresponding scalar product.

We consider the following linear wave propagation problem in Laplace domain: for $f_\delta \in \mathcal{H}$ given, find $u_\delta \in V_\delta$ such that, for all $v \in V_\delta$,

$$(2.6) \qquad s^2(u_\delta, v)_{\mathcal{H}} + (G_\delta u_\delta, G_\delta v)_G = (f_\delta, v)_{\mathcal{H}},$$

where $s = i\omega + \eta$ with $\eta > 0$ and $\omega \in \mathbb{R}$. By application of the Lax-Milgram theorem one can show that this problem admits a unique solution in $V_\delta$. Moreover, choosing $v = s u_\delta$ as a test function in (2.6) one can show the following stability estimate

$$(2.7) \qquad \|s u_\delta\|_{\mathcal{H}}^2 + \|G_\delta u_\delta\|_{G}^2 \le \eta^{-2} \|f_\delta\|_{\mathcal{H}}^2.$$

Note that, such estimate can be used to derive estimates in time domain using the inverse Fourier transform. We refer the reader to [24] or [25] for more details.

#### 2.3.1. Weak convergence towards the limit problem.

**LEMMA 2.4.** Assume that there exists $f_0 \in \mathcal{H}$ such that $f_\delta \underset{\mathcal{H}}{\to} f_0$, then, up to a subsequence,

$$(2.8) \qquad u_\delta \underset{\mathcal{H}}{\to} u_0, \quad G_\delta u_\delta \underset{G}{\to} g_0,$$

where $(u_0, g_0) \in \text{Ker } G_0 \times \text{Ker } G_0^*$ satisfies, for all $v \in \text{Ker } G_0 \cap D(G_1)$,

$$(2.9) \qquad s^2(u_0, v)_{\mathcal{H}} + (g_0, G_1 v)_G = (f_0, v)_{\mathcal{H}}.$$

*Proof.* Since $f_\delta$ weakly converges in $\mathcal{H}$ it is uniformly bounded in $\mathcal{H}$ hence, from (2.7), there exists $C > 0$ independent of $\delta$ such that

$$(2.10) \qquad \|u_\delta\|_{\mathcal{H}}^2 + \|G_\delta u_\delta\|_{G}^2 \le C.$$

From (2.5) we deduce that there exists $u_0 \in \text{Ker } G_0$ such that, up to a subsequence,

$$u_\delta \underset{\mathcal{H}}{\to} u_0.$$

For that subsequence, we choose $v \in D(G_0) \cap D(G_1)$ in (2.6), we have, setting $g_\delta = G_\delta u_\delta$,

$$(2.11) \qquad (g_\delta, G_0 v)_G = \delta(f_\delta, v)_{\mathcal{H}} - \delta s^2(u_\delta, v)_{\mathcal{H}} - \delta(g_\delta, G_1 v)_G.$$

Estimate (2.10) shows that $g_\delta$ is uniformly bounded in $G$, hence, using (2.11) we can apply Theorem 2.3: there exists $g_0 \in \text{Ker } G_0^*$ such that, again, up to a subsequence,

$$g_\delta \underset{G}{\to} g_0.$$
Proof. The following equivalent norm in $V_0$ can be defined: for all $u \in V_0$,

$$ |||u|||^2 = \|su\|_{\mathcal{H}}^2 + \|\overline{Q_0G_1}u\|_{\mathcal{G}}^2 + \frac{|s|^2}{\eta} \Re e(b(u, u)). $$

Using $su_\delta$ and $su_0$ as test functions in (2.20) and (2.24) respectively gives

$$ \begin{cases} |||u_\delta|||^2 + \|(I - Q_0)G_\delta u_\delta\|_{\mathcal{G}}^2 = \frac{1}{\eta} \Re e(\ell_\delta(su_\delta)), \\ |||u_0|||^2 = \frac{1}{\eta} \Re e(\ell_0(su_0)). \end{cases} $$

The first equality above is obtained using,

$$ \|G_{\delta}u_{\delta}\|_{\mathcal{G}}^{2} = \|Q_{0}G_{\delta}u_{\delta}\|_{\mathcal{G}}^{2} + \|(I - Q_{0})G_{\delta}u_{\delta}\|_{\mathcal{G}}^{2} = \|\overline{Q_{0}G_{1}}u_{\delta}\|_{\mathcal{G}}^{2} + \|(I - Q_{0})G_{\delta}u_{\delta}\|_{\mathcal{G}}^{2}. $$

Since $u_\delta$ converges weakly to $u_0$ in $V_0$ (Theorem 2.6) and since, by assumption, $\Re e(\ell_\delta(su_\delta))$ converges to $\Re e(\ell_0(su_0))$ we have

$$ \lim_{\delta \to 0} |||u_\delta|||^2 \le \lim_{\delta \to 0} \frac{1}{\eta} \Re e(\ell_\delta(su_\delta)) = \frac{1}{\eta} \Re e(\ell_0(su_0)) = |||u_0|||^2. $$

However, thanks to the weak convergence property in $V_0$ as well as the equivalence of the norm $||| \cdot |||$ in this space, we have $|||u_0|||^2 \le \liminf_{\delta \to 0} |||u_\delta|||^2$. This shows that

$$ |||u_\delta||| \xrightarrow{\delta \to 0} |||u_0|||, $$

which concludes the proof since, weak convergence and convergence of the norm imply strong convergence. □

**2.3. Scale separations.** In this section two definitions are introduced (strong and strict scale separations). They correspond to simplifying assumptions on $G_0$ and $G_1$ in a formal sequence of two-scale operators. Note that both notions are illustrated Section 3.

**2.3.1. Strong scale separation.**

**DEFINITION 2.9.** A formal sequence of two-scale operators $G_\delta$ satisfies the strong scale separation property when

$$ (2.27) \qquad D(G_0) \cap D(G_1) \text{ dense in } (D(G_0); \| \cdot \|_{\mathcal{H}} + \|G_0 \cdot \|_{\mathcal{G}}) $$

and

$$ \mathrm{Im} G_0 \perp \mathrm{Im} G_1, $$

where the orthogonality holds for the scalar product in $\mathcal{G}$.

**THEOREM 2.10.** Any formal sequence of two-scale operators $G_\delta$ that satisfies the strong scale separation property is a sequence of two-scale operators satisfying the weak scale separation property, $Q_0G_1 = G_1$ and

$$ \left\{ \begin{array}{l} \mathcal{V}_\delta = D(G_0) \cap D(G_1), \\ \mathcal{V}_0 = D(G_1), \\ \mathcal{R}_0 = \mathrm{Ker} G_1 \cap \mathrm{Ker} G_0, \end{array} \right. $$

and (2.13) holds.
Finally, we choose $v \in \text{Ker } G_0 \cap \mathcal{D}(G_1)$ in (2.6) and pass to the limit $\delta \to 0$,

$$s^2(u_\delta, v)_H + (G_\delta u_\delta, G_1 v)_G = (f_\delta, v)_H \Rightarrow s^2(u_0, v)_H + (g_0, G_1 v)_G = (f_0, v)_H$$

this concludes the proof of the lemma.

□

We now recall the definition of the orthogonal projection operator into the kernel of $G_0^*$. It is denoted $Q_0$ and is given by,

$$Q_0 \in \mathcal{L}(\mathcal{G}), \quad \text{Im } Q_0 = \text{Ker } G_0^*, \quad Q_0^2 = Q_0, \quad (g-Q_0g,h)_\mathcal{G} = 0 \quad \forall (g,h) \in \mathcal{G} \times \text{Ker } G_0^*.$$

Note that $Q_0$ is a self-adjoint operator.

A first motivation for introducing the operator $Q_0$ comes from the formal analysis in Section 1.2. Indeed, Equation (1.6) shows that only the operator $Q_0G_1$ appears. However, $Q_0G_1$ is not necessarily closed. This motivates seeking the limit solution in the space

$$V_0 = D(\overline{Q_0G_1}).$$

Here, the closure of the operator is introduced so that $V_0$ is a Hilbert space when equipped with the graph norm and the corresponding scalar product. However, for this construction, it is necessary that $Q_0G_1$ is closable. This will be assumed in the results stated below.

A second motivation concerns the characterization of $g_0$ in Lemma 2.4. The result given in Lemma 2.4 is not very informative, since it is necessary to relate $g_0$ to $u_0$ in some way. However, one can observe that

$$Q_0G_\delta u_\delta \underset{G}{\to} Q_0g_0 = g_0,$$

and $Q_0G_\delta v = \delta^{-1}Q_0G_0v + Q_0G_1v = Q_0G_1v$ for any $v \in D(G_0) \cap D(G_1)$. Hence, one expects that $Q_0G_\delta$ coincides with $Q_0G_1$ on some subspace. These observations are formalized in the next lemma.

LEMMA 2.5. Let $\{G_\delta\}$ be a sequence of two-scale operators and assume that $Q_0G_1$ is closable, then

$$(2.12) \qquad Q_0 G_\delta \subset \overline{Q_0 G_1}.$$

With Lemma 2.4 and 2.5, one can deduce the following convergence theorem.

THEOREM 2.6. Assume that $Q_0G_1$ is closable and that there exists $f_0 \in \mathcal{H}$ such that $f_\delta \underset{\mathcal{H}}{\to} f_0$, then, up to a subsequence,

$$(2.13) \qquad u_\delta \underset{V_0}{\to} u_0,$$

where $u_0 \in \text{Ker } G_0 \cap V_0$, satisfies, for all $v \in \text{Ker } G_0 \cap D(G_1)$,

$$(2.14) \qquad s^2(u_0, v)_H + (\overline{Q_0G_1} u_0, \overline{Q_0G_1} v)_G = (f_0, v)_H.$$
Note that when the strong scale separation holds the problem (2.20) under study reads: find $u_\delta \in D(G_0) \cap D(G_1)$ such that, for all $v \in D(G_0) \cap D(G_1)$,

$$ (2.28) \qquad s^2(u_\delta, v)_H + \delta^{-2}(G_0 u_\delta, G_0 v)_G + (G_1 u_\delta, G_1 v)_G + sb(u_\delta, v) = \ell_\delta(v), $$

whereas the limit problem (2.24) simplifies to: find $u_0 \in \text{Ker } G_0 \cap D(G_1)$, such that, for all $v \in \text{Ker } G_0 \cap D(G_1)$,

$$ (2.29) \qquad s^2(u_0, v)_H + (G_1 u_0, G_1 v)_G + sb(u_0, v) = \ell_0(v). $$

Both problems (2.28) and (2.29) have a unique solution by application of the Lax-Milgram theorem.

### 2.3.2. Strict scale separation.

DEFINITION 2.11. A sequence of formal two-scale operators $G_\delta$ satisfies the strict scale separation property when $\text{Im } G_0 \perp \text{Im } G_1$ and $\text{Im } G_0^* \perp \text{Im } G_1^*$, where the orthogonality holds for the scalar product in $G$ and $H$ respectively.

THEOREM 2.12. Sequences of formal two-scale operators $G_\delta$ satisfying the strict scale separation property satisfy the strong scale separation property and $G_1 P_0 = G_1$.

In Theorem 2.12 above, $P_0$ is the orthogonal projection operator in the kernel of $G_0$. It is the self-adjoint operator defined by,

$$ P_0 \in \mathcal{L}(H), \quad P_0^2 = P_0 \quad (u - P_0 u, v)_H = 0 \quad \forall (u, v) \in H \times \text{Ker } G_0. $$
Proof. Writing $I = (I - Q_0) + Q_0$ and with the orthogonality property of $Q_0$ we have

$$ \|u_{\delta}\|_{\mathcal{H}}^{2} + \|\overline{Q_{0}G_{1}} u_{\delta}\|_{\mathcal{G}}^{2} \le \|u_{\delta}\|_{\mathcal{H}}^{2} + \|Q_{0}G_{\delta} u_{\delta}\|_{\mathcal{G}}^{2} + \|(I - Q_{0}) G_{\delta} u_{\delta}\|_{\mathcal{G}}^{2} \le C. $$

Hence, $u_{\delta}$ is uniformly bounded in $\mathcal{V}_0$ and the weak limit in (2.8) also holds in $\mathcal{V}_0$. We can now characterize $g_0$ in (2.8). We have, thanks to Lemma 2.5,

$$ Q_{0}g_{\delta} = Q_{0}G_{\delta}u_{\delta} = \overline{Q_{0}G_{1}}u_{\delta} \Rightarrow g_{0} = Q_{0}g_{0} = \overline{Q_{0}G_{1}}u_{0}. $$

Equation (2.14) is obtained, first, by using (2.9), second by writing that, for all $v \in \text{Ker } G_0 \cap \mathcal{D}(G_1)$, we have

$$ (g_0, G_1 v)_G = (g_0, Q_0 G_1 v)_G = (g_0, \overline{Q_0 G_1} v)_G = (\overline{Q_0 G_1} u_0, \overline{Q_0 G_1} v)_G. \quad \square $$

Remark 2.7. From Lemma 2.5, since $D(G_{\delta}) = D(Q_0 G_{\delta})$ and $\mathcal{V}_0 = D(\overline{Q_0 G_1})$, one deduces that $\mathcal{V}_{\delta} \subset \mathcal{V}_0$. Moreover as a consequence of the following inequality

$$ \forall v \in \mathcal{V}_{\delta} \quad \|G_{\delta}v\|_{G}^{2} = \|Q_{0}G_{\delta}v\|_{G}^{2} + \|(I - Q_{0})G_{\delta}v\|_{G}^{2} \ge \|Q_{0}G_{\delta}v\|_{G}^{2} = \|\overline{Q_{0}G_{1}}v\|_{G}^{2}, $$

the injection of $\mathcal{V}_{\delta}$ into $\mathcal{V}_0$ is continuous.

Finally, observe that the limit problem (2.14) is not symmetric since the space of test functions is smaller than the solution space (i.e. the space in which the limit solution belongs). This prevent to conclude that the limit solution is unique. One can circumvent this problem with a final assumption.

COROLLARY 2.8. When

$$ (2.15) \qquad \text{Ker } G_0 \cap D(G_1) \text{ is dense in } (\text{Ker } G_0 \cap \mathcal{V}_0; \| \cdot \|_{\mathcal{V}_0}), $$

then $u_0$ is uniquely defined by (2.14).

Proof. When (2.15) holds, density ensures that $u_0$ satisfies (2.14) for every test function in $\text{Ker } G_0 \cap \mathcal{V}_0$. One can then conclude existence and uniqueness of the solution using the Lax-Milgram theorem. In this case, every subsequence converges to the unique solution of problem (2.14). $\square$

2.3.2. Quasi-static solutions and propagating solutions. We introduce the space $\mathcal{R}_0$ related to the kernel of $\overline{Q_0 G_1}$. More precisely, we define

$$ (2.16) \qquad \mathcal{R}_0 := \text{Ker } \overline{Q_0 G_1} \cap \text{Ker } G_0. $$

Since by definition $\overline{Q_0 G_1}$ is closed and densely defined in $\mathcal{H}$ its kernel is closed in $\mathcal{H}$. It is also closed in $\mathcal{V}_0$ since, for all $r \in \text{Ker } \overline{Q_0 G_1}$ we have $\|r\|_{\mathcal{V}_0} = \|r\|_{\mathcal{H}}$. We deduce then that the space $\mathcal{R}_0$ is a closed subspace of $\text{Ker } G_0 \cap \mathcal{V}_0$. We can therefore define $\mathcal{S}_0$ as the orthogonal complement of $\mathcal{R}_0$ in that space. We have

$$ (2.17) \qquad \text{Ker } G_0 \cap \mathcal{V}_0 = \mathcal{R}_0 \oplus \mathcal{S}_0, $$
**3. Applications.** For illustration, two examples in elastic wave propagation are discussed, a problem in piezo-electricity. In this section, $\Omega \subset \mathbb{R}^3$ is a Lipschitz bounded domain. The space $L^2(\Omega; \mathbf{C})$ is the space of second order symmetric tensor equipped with the scalar product $(\cdot, \cdot)_{L^2(\Omega; \mathbf{C})} = (\mathbf{C} \cdot, \cdot)_{L^2(\Omega)^{3 \times 3}}$. where $\mathbf{C}$ is a space dependent, uniformly coercive, elasticity tensor. Below, $u$ is a vector field, $\varepsilon(u)$ its symmetrized gradient.

**3.1. Nearly-incompressible elastodynamics.** After non-dimensionnalization the problems of wave propagation in nearly incompressible reads: for $f \in L^2(\Omega)^3$ given, find $u_\delta \in H^1(\Omega)^3$ such that, for all $v \in H^1(\Omega)^3$,

$$s^2 \int_{\Omega} \rho u_{\delta} \cdot \bar{v} \, dx + \delta^{-2} \int_{\Omega} \operatorname{div} u_{\delta} \operatorname{div} \bar{v} \, dx + \int_{\Omega} \mathbf{C} \varepsilon(u_{\delta}) : \varepsilon(\bar{v}) \, dx = \int_{\Omega} \rho f \cdot \bar{v} \, dx.$$

We define $\mathcal{H} = L^2(\Omega)^3$ equipped with the scalar product $(\cdot, \cdot)_{\mathcal{H}} = (\rho \cdot, \cdot)_{L^2(\Omega)^3}$. We also define $\mathcal{G} = L^2(\Omega) \times L^2(\Omega; \mathbf{C})$. We introduce $G_0$ and $G_1$, closed, densely defined operators from $\mathcal{H}$ to $\mathcal{G}$,

$$D(G_0) = H(\operatorname{div}, \Omega), \quad G_0 u = \begin{pmatrix} \operatorname{div} u \\ 0 \end{pmatrix}, \quad D(G_1) = H^1(\Omega)^3, \quad G_1 u = \begin{pmatrix} 0 \\ \varepsilon(u) \end{pmatrix}.$$

Note that $H(\operatorname{div}, \Omega)$ is the space of function with square integrable divergence. The operators $\delta^{-1}G_0 + G_1$ constitute a sequence of formal two-scale operators. It is easy to verify that $\operatorname{Im} G_0 \perp \operatorname{Im} G_1$. Moreover, from [10] we have that $D(G_0) \cap D(G_1) = H^1(\Omega)^3$ is dense in $D(G_0) = H(\operatorname{div}, \Omega)$. We have thus verified that the strong scale property holds (hence in particular $\mathcal{V}_0 = D(G_1)$). The limit problem is given by: find $u_0 \in \mathcal{V}_0^0$ such that, for all $v \in \mathcal{V}_0^0$,

$$s^2(u_0, v) + (G_1 u_0, G_1 v)_{\mathcal{G}} = s^2(u_0, v) + (\varepsilon(u_0), \varepsilon(v))_{L^2(\Omega; \Sigma)} = (f, v)_{\mathcal{H}},$$

where

$$\mathcal{V}_0^0 = \operatorname{Ker} G_0 \cap D(\overline{G_0 G_1}) = \operatorname{Ker} G_0 \cap D(G_1) = \{u \in H^1(\Omega)^3 | \operatorname{div} u = 0\}.$$

We also have

$$\mathcal{R}_0 = \operatorname{Ker} G_1 \cap \operatorname{Ker} G_0 = \{u \in H^1(\Omega)^3 | \varepsilon(u) = 0\},$$

namely, $\mathcal{R}_0$ is the finite dimensional space of rigid body motions.

**3.2. Elastodynamics in plates.** We are now concerned with the elastic wave propagation in plates with thicknesses – in the $x_3$-direction – proportional to $\delta$. After nondimensionnalisation, the elastodynamics equations are written in a fixed domain $\Omega = S \times [-1, 1]$ where $S \subset \mathbb{R}^2$ is a Lipschitz bounded domain. For simplicity, we consider that the problem is isotropic, i.e.,

$$\forall \varepsilon \in \Sigma, \quad \mathbf{C}\varepsilon = \lambda \sum_{j=1}^{3} \varepsilon_{jj}I + 2\mu\varepsilon,$$

where $I$ is the identity tensor in $\Sigma$ and where $\lambda$ and $\mu$ are the space dependent Lamé parameters of the medium. The problem to solve reads: for $f \in L^2(\Omega)^3$ given, find
where the decomposition is orthogonal with respect to the scalar product in $V_0$, more precisely, we have

$$ \forall (r, s) \in \mathcal{R}_0 \times \mathcal{S}_0, \quad 0 = (r, s)_{V_0} = (r, s)_{\mathcal{H}} + (\overline{Q_0 G_1} r, \overline{Q_0 G_1} s)_{\mathcal{G}} = (r, s)_{\mathcal{H}}, $$

hence $\mathcal{R}_0$ and $\mathcal{S}_0$ are also orthogonal with respect to the scalar product in $\mathcal{H}$. The space $\mathcal{R}_0$ is called the space of quasi-static solutions whereas the space $\mathcal{S}_0$ is called the space of propagating solutions. This terminology is justified below. For all $u_0$ solution to (2.14) we have

$$ u_0 = r_0 + s_0 $$

where $r_0 \in \mathcal{R}_0$ and $s_0 \in \mathcal{S}_0$. When (2.15) holds, they are solutions to

$$ \left\{ \begin{array}{ll} \forall r \in \mathcal{R}_0, & s^2(r_0, r)_{\mathcal{H}} = (f_0, r)_{\mathcal{H}}, \\ \\ \forall v \in \mathcal{S}_0, & s^2(s_0, v)_{\mathcal{H}} + (\overline{Q_0 G_1} s_0, \overline{Q_0 G_1} v)_{\mathcal{G}} = (f_0, v)_{\mathcal{H}}. \end{array} \right. $$

Up to a factor $s^2$, $r_0$ is the projection of $f_0$ on the space $\mathcal{R}_0$ whereas $s_0$ satisfies an abstract wave propagation problem similar to (2.14).

### 2.3.3. Extensions: Lower order terms and more general source term.

Several generalizations of the results given in the previous section can be studied. We focus here on the following wave propagation problem: for $\ell_\delta \in \mathcal{L}(V_0, \mathbb{C})$ given, find $u_\delta \in V_\delta$ such that, for all $v \in V_\delta$,

$$ (2.18) \qquad s^2(u_\delta, v)_{\mathcal{H}} + (G_\delta u_\delta, G_\delta v)_{\mathcal{G}} + sb(u_\delta, v) = \ell_\delta(v). $$

This problem involves, on the one hand, a sesquilinear form $b(\cdot, \cdot)$, that is continuous and positive in $V_0$ (and therefore in $V_\delta$ thanks to Remark 2.7): for all $u \in V_0$ and $v \in V_0$,

$$ (2.19) \qquad |b(u, v)| \le C_b \|u\|_{V_0} \|v\|_{V_0}, \quad \Re(e(b(u, u))) \ge 0, $$

and, on the other hand, a sequence of anti-linear form $\ell_\delta \in \mathcal{L}(V_0, \mathbb{C})$ such that,

$$ (2.20) \qquad \forall v \in V_0, \quad |\ell_\delta(v)| \le c_{\ell,\delta} \|v\|_{\mathcal{H}} + C_{\ell,\delta} \|\overline{Q_0 G_1} v\|_{\mathcal{G}}, $$

for some $c_{\ell,\delta} > 0$ and $C_{\ell,\delta} > 0$ uniformly bounded with respect to $\delta$. Solutions to (2.18) satisfy the following estimate (it is proven using $v = su$ as a test function),

$$ (2.21) \qquad \|s u_\delta\|_{\mathcal{H}}^2 + \|G_\delta u_\delta\|_{\mathcal{G}}^2 \le \eta^{-2} (c_{\ell,\delta} + |s| C_{\ell,\delta})^2. $$

The sesquilinear form $b$ allows to take into account problems with dissipation, viscosity and/or couplings (see Section 4.3 on piezoelectricity for an application example). The anti-linear form $\ell_\delta$ is introduced to take into account problems with boundary data for instance. Both $\ell_\delta$ and $b$ may depend on $s$ (as well as $c_{\ell,\delta}$, $C_{\ell,\delta}$ and $C_b$) in the case of equations obtained from the Laplace transform of time domain problems. We omit this dependence to simplify the reading. Theorem 2.6 can be extended to this case rather straightforwardly (the proof of the result below is therefore omitted).
$u_{\delta} \in H^1(\Omega)^3$ such that, for all $v \in H^1(\Omega)^3$,

$$s^2 \int_{\Omega} \rho u_{\delta} \cdot \bar{v} \, dx + \int_{\Omega} \mathbf{C} (\delta^{-1} \varepsilon_3(u) + \varepsilon_{\sigma}(u)) : (\delta^{-1} \varepsilon_3(\bar{v}) + \varepsilon_{\sigma}(\bar{v})) \, dx = \int_{\Omega} \rho f \cdot \bar{v} \, dx.$$

where

$$\varepsilon_{\sigma}(u) = \frac{1}{2} \begin{pmatrix} 2\partial_1 u_1 & \partial_1 u_2 + \partial_2 u_1 & \partial_1 u_3 \\ \partial_1 u_2 + \partial_2 u_1 & 2\partial_2 u_2 & \partial_2 u_3 \\ \partial_1 u_3 & \partial_3 u_3 & 0 \end{pmatrix},$$

and

$$\varepsilon_3(u) = \frac{1}{2} \begin{pmatrix} 0 & 0 & \partial_3 u_1 \\ 0 & 0 & \partial_3 u_2 \\ \partial_3 u_1 & \partial_3 u_2 & 2\partial_3 u_3 \end{pmatrix}.$$

Note that $\varepsilon(u) = \varepsilon_3(u) + \varepsilon_{\tau}(u)$. As in Section 3.1, in order to write these equations in the abstract setting the space $\mathcal{H}$ is defined by $\mathcal{H} = L^2(\Omega)^3$ and $\mathcal{G}$ by $\mathcal{G} = L^2(\Omega; \Sigma)$. Moreover,

$$D(G_0) = \{u \in L^2(\Omega)^3 \mid \varepsilon_3(u) \in L^2(\Omega; \Sigma)\} \quad G_0 u = \varepsilon_3(u),$$

and

$$D(G_1) = \{u \in L^2(\Omega)^3 \mid \varepsilon_{\tau}(u) \in L^2(\Omega; \Sigma)\} \quad G_1 u = \varepsilon_{\tau}(u),$$

One can show that $\delta^{-1}G_0 + G_1$ is closable (it is in fact closed), densely defined, and

$$G_{\delta} = \overline{\delta^{-1}G_{0} + G_{1}}$$

is defined by

$$D(G_{\delta}) = \{u \in L^2(\Omega)^3 \mid \varepsilon(u) \in L^2(\Omega; \Sigma)\} = H^1(\Omega)^3,$$

and, for all $u \in H^1(\Omega)^3$,

$$G_{\delta}u = \delta^{-1}\varepsilon_3(u) + \varepsilon_{\sigma}(u).$$

Note that the Korn's inequality has been used to deduce that $D(G_{\delta}) = H^1(\Omega)^3$. The problem now reads: find $u_{\delta} \in H^1(\Omega)^3$ such that, for all $v \in H^1(\Omega)^3$,

$$s^2(u, v)_{\mathcal{H}} + (G_{\delta}u, G_{\delta}v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}.$$

To check that $G_{\delta}$ is a sequence of two-scale operators the adjoints $G_0$ and $G_1$ have to be computed. In order to define these adjoints, the following divergence operators are introduced

$$\forall \varepsilon \in L^2(\Omega; \mathbf{C}), \quad \mathbf{Div}_3 = \sum_{j=1}^{3} \partial_3 \varepsilon_{j3} \mathbf{e}_j, \quad \mathbf{Div}_{\sigma} = \mathbf{Div} - \mathbf{Div}_3,$$

where $\mathbf{e}_j$ is the $j$th element of the canonical basis and $\mathbf{Div}$ is the standard divergence operator of a second order tensor. The boundary of $\Omega$ is decomposed using the following two disjoints subdomains

$$\Gamma_S = S \times \{-1, 1\}, \quad \Gamma_\sigma = \partial S \times (-1, -1).$$

The unitary outward normal on $\Gamma_{\sigma}$ is denoted $\mathbf{n}_{\sigma}$. With these notations, the adjoints are given by

$$\left\{ \begin{array}{l} D(G_0^*) = \{\varepsilon \in L^2(\Omega; \mathbf{C}) \mid \mathbf{Div}_3(\mathbf{C}\varepsilon) \in L^2(\Omega)^3, (\mathbf{C}\varepsilon)\mathbf{e}_3 = 0 \text{ on } \Gamma_S\}, \\ D(G_1^*) = \{\varepsilon \in L^2(\Omega; \mathbf{C}) \mid \mathbf{Div}_{\sigma}(\mathbf{C}\varepsilon) \in L^2(\Omega)^3, (\mathbf{C}\varepsilon)\mathbf{n}_{\sigma} = 0 \text{ on } \Gamma_{\sigma}\}. \end{array} \right.$$
**THEOREM 2.9.** Assume that $Q_0G_1$ is closable and that there exists an anti-linear form $\ell_0 \in \mathcal{L}(V_0, \mathbb{C})$ such that

$$ \forall v \in \ker G_0 \cap D(G_1), \quad \ell_\delta(v) \underset{\delta \to 0}{\longrightarrow} \ell_0(v) $$

then, up to a subsequence,

$$ u_\delta \underset{V_0}{\longrightarrow} u_0. $$

where $u_0 \in \ker G_0 \cap V_0$, satisfies, for all $v \in \ker G_0 \cap D(G_1)$,

$$ (2.22) \qquad s^2(u_0, v)_{\mathcal{H}} + (\overline{Q_0G_1} u_0, \overline{Q_0G_1} v)_{\mathcal{G}} + sb(u_0, v) = \ell_0(v). $$

When $\ker G_0 \cap D(G_1)$ is dense in $\ker G_0 \cap V_0$, then $u_0$ is uniquely defined by (2.22).

### 2.3.4. Strong convergence.

**THEOREM 2.10.** Assume that there exists an anti-linear form $\ell_0 \in \mathcal{L}(V_0, \mathbb{C})$, such that

$$ (2.23) \qquad \|\ell_\delta - \ell_0\|_{\mathcal{L}(V_0, \mathbb{C})} \underset{\delta \to 0}{\longrightarrow} 0, $$

and assume that $Q_0G_1$ is closable and $\ker G_0 \cap D(G_1)$ dense in $\ker G_0 \cap V_0$, then, the sequence of solutions $\{u_\delta\} \subset V_0$ to (2.18), satisfies

$$ (2.24) \qquad u_\delta \underset{V_0}{\longrightarrow} u_0, $$

where $u_0$ is the unique solution to (2.22).

*Proof.* The following equivalent norm in $V_0$ can be defined,

$$ \forall u \in V_0, \quad \|u\|^2 = \|su\|_{\mathcal{H}}^2 + \|\overline{Q_0G_1}u\|_{\mathcal{G}}^2 + \frac{|s|^2}{\eta} \Re(e(b(u, u))). $$

Using $su_\delta$ and $su_0$ as test functions in (2.18) and (2.22) respectively gives

$$ \left\{ \begin{array}{l} |||u_\delta|||^2 + \|(I - Q_0)G_\delta u_\delta\|_{\mathcal{G}}^2 = \frac{1}{\eta} \Re(e(\ell_\delta(su_\delta))), \\ |||u_0|||^2 = \frac{1}{\eta} \Re(e(\ell_0(su_0))). \end{array} \right. $$

Note that, the first equality above is obtained using,

$$ \|G_\delta u_\delta\|_{\mathcal{G}}^2 = \|Q_0 G_\delta u_\delta\|_{\mathcal{G}}^2 + \|(I - Q_0)G_\delta u_\delta\|_{\mathcal{G}}^2 = \|\overline{Q_0 G_1} u_\delta\|_{\mathcal{G}}^2 + \|(I - Q_0)G_\delta u_\delta\|_{\mathcal{G}}^2. $$

Since $u_\delta$ converges weakly to $u_0$ in $V_0$ (Theorem 2.9) we deduce, from (2.23), that $\Re(\ell_\delta(su_\delta))$ converges to $\Re(\ell_0(su_0))$, hence

$$ \limsup_{\delta \to 0} |||u_\delta|||^2 \le \lim_{\delta \to 0} \frac{1}{\eta} \Re(e(\ell_\delta(su_\delta))) = \frac{1}{\eta} \Re(e(\ell_0(su_0))) = |||u_0|||^2. $$

However, thanks to the weak convergence property in $V_0$ as well as the equivalence of the norm $||| \cdot |||$ in this space, we have $|||u_0|||^2 \le \liminf_{\delta \to 0} |||u_\delta|||^2$. This shows that

$$ |||u_\delta||| \underset{\delta \to 0}{\longrightarrow} |||u_0|||, $$

which concludes the proof since, weak convergence and norm convergence imply strong convergence. □
and

$$G_0^*\epsilon = -\rho^{-1}\mathbf{Div}_3(\mathbf{C}\epsilon), \quad G_1^*\epsilon = -\rho^{-1}\mathbf{Div}_\sigma(\mathbf{C}\epsilon).$$

It can be verified that $D(G_0) \cap D(G_1)$ is dense in $D(G_0)$ and $D(G_0^*) \cap D(G_1^*)$ is dense in $D(G_0^*)$. The proof simply amounts to use the density of smooth functions in $H^1$-spaces. Moreover,

$$\text{Ker } G_0^* = \{\epsilon \in L^2(\Omega; \mathbf{C}) \mid (\mathbf{C}\epsilon)\mathbf{e}_3 = 0 \text{ in } \Omega\},$$

and for isotropic media (here the isotropy is used to obtain a simple expression) one can check that the orthogonal projection operator on $\text{Ker } G_0^*$ is given by

$$\forall \epsilon \in L^2(\Omega; \mathbf{C}), \quad Q_0\epsilon = \begin{pmatrix} \epsilon_{11} & \epsilon_{12} & 0 \\ \epsilon_{12} & \epsilon_{22} & 0 \\ 0 & 0 & \frac{-\lambda(\epsilon_{11} + \epsilon_{22})}{\lambda + 2\mu} \end{pmatrix},$$

After a straightforward formal computation of $Q_0G_1u$, one can then show that, setting

$$\mathcal{V}_0^0 = D(\overline{Q_0G_1}) \cap \text{Ker } G_0 = \{u \in L^2(\Omega)^3 \mid \partial_3 u = 0, u_1 \in H^1(\Omega), u_2 \in H^1(\Omega)\},$$

the limit problem is given by: find $u_0 \in \mathcal{V}_0^0$ such that for all $v \in V_0^0$

$$(3.1) \qquad s^2(u_0, v) + (\alpha\lambda\text{div}_\sigma u_0, \text{div}_\sigma v) + 2(\mu\epsilon_\sigma(u_0), \epsilon_\sigma(v))_{L^2(\Omega)^{3\times3}} = (f, v)_{\mathcal{H}},$$

where $\text{div}_\sigma v = \partial_1 v_1 + \partial_2 v_2$, $\alpha = 2\mu/(\lambda + 2\mu)$. It should be noted here that $u_3$ is not sought in $H^1(\Omega)$. Even more remarkable is the fact that the space

$$\mathcal{R}_0 := \text{Ker } \overline{Q_0G_1} \cap \text{Ker } G_0 = \{u \in L^2(\Omega)^3 \mid u_1 = 0, u_2 = 0, \partial_3 u_3 = 0\},$$

is of infinite dimension and corresponds to functions that are polarized only in the $x_3$-direction and independent of $x_3$.

**3.3. Piezoelectricity.** Piezoelectric equations coupled electromagnetic fields $(E, H)$ and velocity fields $v$ (where $v = \partial_t u$) thanks to a third order tensor $\mathbf{d}$ namely, the piezoelectric tensor. This tensor maps vector fields to tensor fields,

$$\forall x \in \Omega, \quad \forall \mathbf{E} \in \mathbb{R}^3, \quad d(x)\mathbf{E} \in \Sigma.$$

The objective of this section is to explain how the quasi-static assumption in piezoelectricity (in particular, the electric field reduces to a gradient) can be justified by the proposed abstract framework. After non-dimensionnalization, using Dichlet boundary conditions on the boundary and assuming that the propagating media is surrounded by a perfectly conducting material we arrive at (again see [13] for more details) the following variational formulation: for $F \in L^2(\Omega)^3$ and $J \in L^2(\Omega)^3$ given, find $(E_\delta, v_\delta) \in H_0(\text{curl}; \Omega) \times H_0^1(\Omega)^3$ such that, for all $(\tilde{\mathbf{E}}, \tilde{\mathbf{v}}) \in H_0(\text{curl}; \Omega_E) \times H_0^1(\Omega)^3$,

$$(3.2) \quad \left\{ \begin{array}{l} s^2(E_\delta, \tilde{E})_{L^2(\Omega)^3} + \delta^{-2}(\text{curl } E_\delta, \text{curl } \tilde{E})_{L^2(\Omega)^3} \\ \qquad + s(\epsilon(v_\delta), \mathbf{d}\tilde{E})_{L^2(\Omega)^3} = (J, \tilde{E})_{L^2(\Omega)^3}, \\[2ex] s^2(v_\delta, \tilde{v})_{L^2(\Omega)^3} + (\epsilon(v_\delta), \epsilon(\tilde{v}))_{L^2(\Omega;\Sigma)} \\ \qquad - s(\epsilon(\tilde{v}), \mathbf{d}E_\delta)_{L^2(\Omega)^3} = (F, \tilde{v})_{L^2(\Omega)^3}. \end{array} \right.$$
**2.4. Scale separations.** In this section two definitions are introduced (strong and strict scale separations). They correspond to simplifying assumptions on $G_0$ and $G_1$ when considering formal sequences of two-scale operators. Note that both notions are illustrated Section 4.

**2.4.1. Strong scale separation.**

**DEFINITION 2.11.** A formal sequence of two-scale operators $G_\delta$ satisfies the strong scale separation property when

$$ (2.25) \qquad D(G_0) \cap D(G_1) \text{ dense in } (D(G_0); \|\cdot\|_{\mathcal{H}} + \|G_0\|_{\mathcal{G}}) $$

and

$$ \mathrm{Im} G_0 \perp \mathrm{Im} G_1, $$

where the orthogonality holds for the scalar product in $\mathcal{G}$.

**THEOREM 2.12.** Any formal sequence of two-scale operators $G_\delta$ that satisfies the strong scale separation property is a sequence of two-scale operators. The operator $\delta^{-1}G_0 + G_1$ is closed, $Q_0G_1 = G_1$ and

$$ \left\{ \begin{array}{l} V_\delta = D(G_0) \cap D(G_1), \\ V_0 = D(G_1), \\ \mathcal{R}_0 = \mathrm{Ker} G_1 \cap \mathrm{Ker} G_0. \end{array} \right. $$

Note that when the strong scale separation holds the problem (2.18) under study reads: find $u_\delta \in D(G_0) \cap D(G_1)$ such that, for all $v \in D(G_0) \cap D(G_1)$,

$$ (2.26) \qquad s^2(u_\delta, v)_\mathcal{H} + \delta^{-2}(G_0 u_\delta, G_0 v)_\mathcal{G} + (G_1 u_\delta, G_1 v)_\mathcal{G} + sb(u_\delta, v) = \ell_\delta(v), $$

whereas the limit problem (2.22) simplifies to: find $u_0 \in \mathrm{Ker} G_0 \cap D(G_1)$, such that, for all $v \in \mathrm{Ker} G_0 \cap D(G_1)$,

$$ (2.27) \qquad s^2(u_0, v)_\mathcal{H} + (G_1 u_0, G_1 v)_\mathcal{G} + sb(u_0, v) = \ell_0(v). $$

**2.4.2. Strict scale separation.**

**DEFINITION 2.13.** A sequence of formal two-scale operators $G_\delta$ satisfies the strict scale separation property when

$$ \mathrm{Im} G_0 \perp \mathrm{Im} G_1 \quad \text{and} \quad \mathrm{Im} G_0^* \perp \mathrm{Im} G_1^*, $$

where the orthogonality holds for the scalar product in $\mathcal{G}$ and $\mathcal{H}$ respectively.

**THEOREM 2.14.** Sequences of formal two-scale operators $G_\delta$ satisfying the strict scale separation property satisfy the strong scale separation property.
The small parameter $\delta$ represents the ratio of velocities of waves in solids and electromagnetic waves. The space $H_0(\text{curl}; \Omega)$ is the space of functions with square integrable rotational and vanishing tangential trace. By setting $\mathcal{H} = L^2(\Omega)^3 \times L^2(\Omega)^3$ and $\mathcal{G} = L^2(\Omega)^3 \times L^2(\Omega; \Sigma)$ and defining

$$D(G_{\delta}) = \{\Phi = \begin{pmatrix} \mathbf{E} \\ \mathbf{v} \end{pmatrix} \in H_0(\text{curl}; \Omega_E) \times H^1(\Omega)^3\}, \quad G_{\delta}\Phi = G_{\delta}\begin{pmatrix} \mathbf{E} \\ \mathbf{v} \end{pmatrix} = \begin{pmatrix} \delta^{-1}\text{curl}\mathbf{E} \\ \varepsilon(\mathbf{v}) \end{pmatrix},$$

problem (3.2) can be written as: for $\Psi = (F, J)^t \in \mathcal{H}$ given, find $\Phi_{\delta} \in D(G_{\delta})$ such that, for all $\tilde{\Phi} \in D(G_{\delta})$,

$$(3.3) \qquad s^2(\Phi_{\delta}, \tilde{\Phi})_{\mathcal{H}} + (G_{\delta}\Phi_{\delta}, G_{\delta}\tilde{\Phi})_{\mathcal{G}} + b(s\Phi_{\delta}, \tilde{\Phi}) = (\Psi, \tilde{\Phi})_{\mathcal{H}},$$

with, for all $\Phi = \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_{\delta})$ and $\tilde{\Phi} = \begin{pmatrix} \tilde{E} \\ \tilde{v} \end{pmatrix} \in D(G_{\delta})$,

$$b(\Phi, \tilde{\Phi}) = (\varepsilon(v), \mathbf{d}\tilde{E})_{L^2(\Omega)^3} - (\varepsilon(\tilde{v}), \mathbf{d}E)_{L^2(\Omega)^3}.$$

In order to show that (3.3) is a two scale propagation problem as defined by (2.20) one needs to show first that $G_{\delta}$ is a sequence of two-scale operators satisfying the weak scale separation property, second that $b$ is a continuous sesquilinear form in $V_0$. The operator $G_0$ and $G_1$ are defined by

$$D(G_0) = H_0(\text{curl}; \Omega) \times L^2(\Omega)^3, \quad \forall \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_0), \quad G_0\begin{pmatrix} E \\ v \end{pmatrix} = \begin{pmatrix} \text{curl} E \\ 0 \end{pmatrix}$$

and

$$D(G_1) = L^2(\Omega)^3 \times H_0^1(\Omega)^3, \quad \forall \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_1), \quad G_1\begin{pmatrix} E \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ \varepsilon(v) \end{pmatrix}.$$

They are closed, densely defined operators. One can verify that the formal sequence of two-scale operators $G^{\delta} = \delta^{-1}G_0 + G_1$ satisfies the strict scale separation property hence the weak scale separation property. In particular $\overline{Q_0G_1} = G_1$ and

$$V_0 = D(\overline{Q_0G_1}) = D(G_1).$$

One can then verify easily that the sesquilinear form $b$ is indeed continuous in $V_0$. The limit problem reads: for $\Psi = (F, J)^t \in \mathcal{H}$ given, find $\Phi_0 = (E_0, v_0)^t \in \text{Ker } G_0 \cap D(G_1)$ such that, for all $\tilde{\Phi} \in \text{Ker } G_0 \cap D(G_1)$,

$$(3.4) \qquad s^2(\Phi_0, \tilde{\Phi})_{\mathcal{H}} + (\varepsilon(v_0), \varepsilon(\tilde{v}))_{L^2(\Omega;\Sigma)} + b(s\Phi_0, \tilde{\Phi}) = (\Psi, \tilde{\Phi})_{\mathcal{H}},$$

where, thanks to [10] (Chapter 1, Theorem 2.7 and Corrolary 2.9),

$$\text{Ker } G_0 = \left\{ \begin{pmatrix} E \\ v \end{pmatrix} \in \mathcal{H} \mid \text{curl } E = 0 \text{ in } \Omega \right\} = \nabla H_{\gamma}^1(\Omega) \times L^2(\Omega)^3.$$

with

$$H_{\gamma}^{1}(\Omega) = \{\varphi \in H^{1}(\Omega)/\mathbb{R} \mid \varphi \text{ is constant along } \partial\Omega\}.$$

Therefore introducing $\varphi_0 \in H_{\gamma}^1(\Omega)$ such that $E_0 = \nabla\varphi_0$ and $u_0 = v_0/s$, the variational formulation of the limit problem can be simplified. It reads: for $(F, J)^t \in \mathcal{H}$ given, find $(\varphi_0, u_0) \in H_{\gamma}^1(\Omega) \times H_0^1(\Omega)^3$ such that, for all, $(\tilde{\varphi}, \tilde{u}) \in H_{\gamma}^1(\Omega) \times H_0^1(\Omega)^3$,

$$\left\{ \begin{array}{l} (\nabla\varphi_0, \nabla\tilde{\varphi})_{L^2(\Omega)^3} + (\varepsilon(u_0), \mathbf{d}\nabla\tilde{\varphi})_{L^2(\Omega)^3} = (s^{-2}J, \tilde{E})_{L^2(\Omega)^3}, \\ s^2(u_0, \tilde{v})_{L^2(\Omega)^3} + (\varepsilon(u_0), \varepsilon(\tilde{v}))_{L^2(\Omega;\Sigma)} - (\varepsilon(\tilde{v}), \mathbf{d}\nabla\varphi_0)_{L^2(\Omega)^3} = (s^{-1}F, \tilde{v})_{L^2(\Omega)^3}. \end{array} \right.$$

These last equations are classically known as the quasi-static piezoelectric equations.
### 3. Application in 1D Periodic homogenization.

This section focuses on 1D periodic homogenization using the shift method from [30]. Its primary goal is to demonstrate an application of the abstract framework introduced in Section 2 and to derive convergence results similar to some of the results given in [30]. Note that the following result is used later in this section.

LEMMA 3.1. Let $G_0$ and $G_1$ be densely defined closed operators from $\mathcal{H}$ to $\mathcal{G}$ and let $Q_0$ be the orthogonal projection operator on $\text{Ker } G_0^*$, then $Q_0 G_1$ is closable if and only if $\text{Ker } G_0^* \cap D(G_1^*)$ is dense in $(\text{Ker } G_0^*; \| \cdot \|_{\mathcal{G}})$.

**Two-scale formulation.** Setting, $\Omega = (0, L)$ and $Y = (0, 1)$, the considered variational formulation reads: find $u_\delta \in H^1(\Omega)$ such that for all $v \in H^1(\Omega)$,

$$ (3.1) \qquad s^2(u_\delta, v)_{L^2(\Omega)} + (\mu_\delta \frac{d}{dx} u_\delta, \frac{d}{dx} v)_{L^2(\Omega)} = (f, v)_{L^2(\Omega)}. $$

The function $\mu_\delta$ is given by

$$ \mu_\delta(x) = \mu_0\left(x, \frac{x}{\delta}\right) $$

where $\mu_0(x, y)$ is a Y-periodic function defined for every $(x, y) \in \Omega \times \mathbb{R}$ and is uniformly bounded by above and by below by strictly positive values. We assume the following weak regularity requirement for $\mu_0$,

$$ (3.2) \qquad \text{a.e. } y \in Y, \quad \forall x \in \Omega, \quad z \to \mu_0(x, y + z) \text{ is continuous at } z = 0. $$

Such assumption is satisfied for every $\mu_0$ that is a piecewise smooth function that may be discontinuous on some finite numbers of lines of the form $x = c$ or $y = c$.

In order to write (3.1) as a problems involving a two-scale operator we follow the work of [30] and [26]. For all $z \in \mathbb{R}$, we introduce the shifted problem: find $u_{\delta,z} \in H^1(\Omega)$ such that for all $v \in H^1(\Omega)$,

$$ (3.3) \qquad s^2(u_{\delta,z}, v)_{L^2(\Omega)} + (\mu_0(x, \frac{x}{\delta} + z)\partial_x u_{\delta,z}, \partial_x v)_{L^2(\Omega)} = (f, v)_{L^2(\Omega)}. $$

Note that (3.1) is recovered by setting $z = 0$. Let $\{z_n\}_{n \in \mathbb{N}} \subset \mathbb{R}$ be a sequence converging towards $z$, we have, setting $e_{\delta,n} = u_{\delta,z_n} - u_{\delta,z}$,

$$ \begin{aligned} (3.4) \qquad s^2(e_{\delta,n}, v)_{L^2(\Omega)} &+ (\mu_0(x, \frac{x}{\delta} + z_n)\partial_x e_{\delta,n}, \partial_x v)_{L^2(\Omega)} \\ & = ((\mu_0(x, \frac{x}{\delta} + z) - \mu_0(x, \frac{x}{\delta} + z_n))\partial_x u_{\delta,z}, \partial_x v)_{L^2(\Omega)}. \end{aligned} $$

Choosing $v = se_{\delta,n}$ as a test function leads to the estimate

$$ \|se_{\delta,n}\|_{L^2(\Omega)}^2 + \|\partial_x e_{\delta,n}\|_{L^2(\Omega)}^2 \le \frac{C}{\eta} \int_{\Omega} |\mu_0(x, \frac{x}{\delta} + z) - \mu_0(x, \frac{x}{\delta} + z + (z_n - z))| \|\partial_x u_{\delta,z}\|^2 dx, $$

where $C$ depends only on the upper and lower bounds of $\mu_0$. Thanks to the regularity Assumption 3.2 one can deduce – using the dominated convergence theorem – that

$$ \|u_{\delta,z_n} - u_{\delta,z}\|_{L^2(\Omega)}^2 + \|\partial_x u_{\delta,z_n} - \partial_x u_{\delta,z}\|_{L^2(\Omega)}^2 \underset{n \to +\infty}{\longrightarrow} 0. $$
**4. Convergence estimates.** As in the proof of Theorem 2.8 the following equivalent norm in $V_0$ is introduced: for all $u \in V_0$,

$$ |||u|||^2 = \|su\|_{\mathcal{H}}^2 + \|\overline{Q_0G_1}u\|_{\mathcal{G}}^2 + \frac{|s|^2}{\eta} \Re(b(u, u)). $$

**4.1. Strict scale separation.**

**THEOREM 4.1.** Let the strict scale separation property holds and assume that the range of $G_0$ is closed, namely there exists $C_p > 0$ such that

$$ (4.1) \qquad \forall u \in \ker G_0^{\perp} \cap D(G_0), \quad \|u\|_{\mathcal{H}} \le C_p \|G_0 u\|_{\mathcal{G}}. $$

Assume, moreover, that there exists an anti-linear form $\ell_0 \in \mathcal{L}(D(G_1))$ such that

$$ \forall v \in \ker G_0 \cap D(G_1), \quad |\ell_\delta(v) - \ell_0(v)| \le c_{\ell,\delta} \|v\|_{\mathcal{H}} + C_{\ell,\delta} \|G_1 v\|_{\mathcal{G}}. $$

Then, the sequence of solutions $\{u_\delta\} \subset D(G_0) \cap D(G_1)$ to (2.28), satisfies

$$ (4.2) \qquad (|||u_\delta - u_0|||^2 + \delta^{-2} \|G_0 u_\delta\|_{\mathcal{G}}^2)^{\frac{1}{2}} \le C_\delta $$

where

$$ C_{\delta} = \frac{1}{\eta} \max(c_{\ell,\delta}, |s|C_{\ell,\delta}, \delta C_0) \quad \text{with} \quad C_0 = |s|C_p(c_{\ell} + C_b \|su_0\|_{D(G_1)}). $$

**Proof.**

**Step 1: Decomposition of the solution.** Since $\ker G_0$ is a closed subspace of $\mathcal{H}$, the solution $u_\delta \in D(G_0) \cap D(G_1)$ to (2.28) can be decomposed as

$$ (4.3) \qquad u_{\delta} = u_{\delta,0} + z_{\delta}, $$

where $u_{\delta,0} \in \ker G_0$ and $z_\delta \in (\ker G_0)^\perp$ and the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. From the strict scale separation property,

$$ (\ker G_0)^\perp = \overline{\mathrm{Im} \, G_0^*} \subset \ker G_1, $$

where the closure is taken for the norm in $\mathcal{H}$. This implies that $z_\delta \in \ker G_1$, and therefore

$$ u_{\delta,0} \in \ker G_0 \cap D(G_1) \quad \text{and} \quad z_\delta \in (\ker G_0)^\perp \cap \ker G_1. $$

**Step 2: Energy identity for the error equation.** The error term

$$ e_{\delta} = u_{\delta,0} - u_0 \in \ker G_0 \cap D(G_1), $$

is introduced. From the decomposition (4.3) it can be deduced that, for all $v \in \ker G_0 \cap D(G_1)$,

$$ (4.4) \qquad s^2(e_\delta, v)_{\mathcal{H}} + (G_1 e_\delta, G_1 v) + b(se_\delta + sz_\delta, v) = \ell_\delta(v) - \ell_0(v). $$
Hence $u_{\delta,z}$ is continuous with respect to $z \in \mathbb{R}$, more precisely

$$u_{\delta,z} \in C^0(\mathbb{R}; H^1(\Omega)).$$

One straightforwardly deduces that $u_{\delta,z}$ is also $Y$-periodic in $z$. We introduce now the function

$$a.e. (x, y) \in \Omega \times Y \quad U_{\delta}(x, y) := u_{\delta, y-x/\delta}(x).$$

Thanks to the continuity of $u_{\delta,z}$ with respect to $z$, one can deduce that $U_{\delta}(x, y)$ belongs to $L^2(\Omega \times Y)$, moreover, for almost all $x \in \Omega$, $U_{\delta}(x, y)$ is continuous and periodic with respect to $y$. By differentiating in the sense of distribution one finds

$$(\partial_x + \delta^{-1}\partial_y) U_{\delta} = (\partial_x u_{\delta,z})_{|z=y-x/\delta} \in L^2(\Omega \times Y).$$

Now, for all $V(x, y)$ in

$$C_{\#}^{\infty}(\bar{\Omega} \times \mathbb{R}) = \{V \in C^{\infty}(\bar{\Omega} \times \mathbb{R}) \mid V(x, y+1) = V(x, y), (x, y) \in \bar{\Omega} \times \mathbb{R}\},$$

one can choose a smooth test function $v_z(x) = V(x, z+x/\delta)$ in (3.3). Obviously $v_z$ is 1-periodic with respect to $z \in \mathbb{R}$ and

$$v_{y-x/\delta}(x) = V(x, y), \quad (\partial_x v_z)|_{z=y-x/\delta} = (\partial_x + \delta^{-1}\partial_y) V(x, y).$$

Integrating (3.3) with respect to $z \in Y$ and using the change of variable $z = y - x/\delta$, we arrive at, for all $V \in C_{\#}^{\infty}(\bar{\Omega} \times \mathbb{R})$,

$$(3.5) \quad s^2(U_{\delta}, V)_{L^2(\Omega \times Y)} + (\mu_0 (\partial_x + \delta^{-1}\partial_y) U_{\delta}, (\partial_x + \delta^{-1}\partial_y) V)_{L^2(\Omega \times Y)} \\ = (f, V)_{L^2(\Omega \times Y)}.$$

One now defines the following Hilbert space

$$H^{1,\delta}(\Omega \times Y) = \{U \in L^2(\Omega \times Y) \mid (\partial_x + \delta^{-1}\partial_y) U \in L^2(\Omega \times Y)\},$$

equipped with the norm: for all $v \in H^{1,\delta}(\Omega \times Y)$,

$$\|v\|_{H^{1,\delta}(\Omega \times Y)}^2 = \|v\|_{L^2(\Omega \times Y)}^2 + \|(\partial_x + \delta^{-1}\partial_y)v\|_{L^2(\Omega \times Y)}^2,$$

then, denoting $C_{\#}^{\infty}(\bar{\Omega} \times Y)$ the restriction of functions in $C_{\#}^{\infty}(\bar{\Omega} \times \mathbb{R})$ to $\bar{\Omega} \times Y$, we define

$$H_{\#}^{1,\delta}(\Omega \times Y) = \overline{C_{\#}^{\infty}(\bar{\Omega} \times Y)}^{\|\cdot\|_{H^{1,\delta}(\Omega \times Y)}}.$$

The two-scale problem finally reads: given $f \in L^2(\Omega)$, finds $U_{\delta} \in H_{\#}^{1,\delta}(\Omega \times Y)$, satisfying (3.5) for all $V \in H_{\#}^{1,\delta}(\Omega \times Y)$.

**The sequence of two-scale operators.** The space $\mathcal{H} = L^2(\Omega \times Y)$ and $\mathcal{G} = L^2(\Omega \times Y)$ are equipped with weighted $L^2$-scalar products with weight 1 and $\mu_0$ respectively. Two unbounded operators $G_0$ and $G_1$ are defined,

$$\left\{ \begin{array}{ll} D(G_0) = \{U \in \mathcal{H} \mid \partial_y U \in \mathcal{G}, U(\cdot, 0) = U(\cdot, 1)\}, & G_0 U = \partial_y U, \\ D(G_1) = \{U \in \mathcal{H} \mid \partial_x U \in \mathcal{G}\}, & G_1 U = \partial_x U. \end{array} \right.$$
Choosing $v = se_\delta$ in (4.4) yields, using the orthogonality of $e_\delta$ and $z_\delta$ in $\mathcal{H}$,

$$ (4.5) \qquad s \|s e_\delta\|_{\mathcal{H}}^2 + \bar{s} \|G_1 e_\delta\|_{\mathcal{G}}^2 + b(se_\delta + sz_\delta, se_\delta) = (s\ell_\delta(e_\delta) - \ell_0(se_\delta)). $$

Moreover, choosing $v = sz_\delta$ in (2.28) yields

$$ (4.6) \qquad s \|s z_\delta\|_{\mathcal{H}}^2 + \bar{s} \delta^{-2} \|G_0 z_\delta\|_{\mathcal{G}}^2 + b(su_{\delta,0} + sz_\delta, sz_\delta) = \ell_\delta(sz_\delta). $$

**Step 3: Error estimate.** The next step is to sum (4.5) and (4.6), to get

$$ (4.7) \qquad \begin{aligned} & |||e_\delta + z_\delta|||^2 + \delta^{-2} \|G_0 z_\delta\|_{\mathcal{G}}^2 \\ & = \frac{1}{\eta} \Re \big( \ell_\delta(se_\delta) - \ell_0(se_\delta) + \ell_\delta.sz_\delta - b(su_0, sz_\delta) \big). \end{aligned} $$

where, due to the properties of $e_\delta$ and $z_\delta$,

$$ |||e_\delta + z_\delta|||^2 = \|s e_\delta\|_{\mathcal{H}}^2 + \|s z_\delta\|_{\mathcal{G}}^2 + \|G_1 e_\delta\|_{\mathcal{G}}^2 + \frac{1}{\eta} \Re \big( b(se_\delta + sz_\delta, se_\delta + sz_\delta) \big). $$

The three terms in the right hand side of Equation (4.7) are now estimated.

$$ (4.8) \qquad \left| \left| \ell_\delta(se_\delta) - \ell_0(se_\delta) \right| \right| \le c_{\ell,\delta} \|se_\delta\|_{\mathcal{H}} + C_{\ell,\delta} |s| \|G_1 e_\delta\|_{\mathcal{G}} $$

moreover, since $z_\delta \in (\text{Ker } G_0)^\perp \cap \text{Ker } G_1$, and since the ranged of $G_0$ is closed,

$$ (4.9) \qquad | \ell_\delta.sz_\delta | \le c_\ell \|sz_\delta\|_{\mathcal{H}} \le c_\ell C_p |s| \|G_0 z_\delta\|_{\mathcal{G}}. $$

Finally, using the continuity property of $b$ and again the fact that the ranged of $G_0$ is closed,

$$ (4.10) \qquad |b(su_0, sz_\delta)| \le C_b \|su_0\|_{D(G_1)} \|sz_\delta\|_{D(G_1)} \le C_b C_p |s| \|su_0\|_{D(G_1)} \|G_0 z_\delta\|_{\mathcal{G}} $$

Summing (4.8), (4.9), (4.10), leads to,

$$ (4.11) \qquad \begin{aligned} \mathcal{E}_\delta &:= |||e_\delta + z_\delta|||^2 + \delta^{-2} \|G_0 z_\delta\|_{\mathcal{G}}^2 \\ &\le \frac{c_{\ell,\delta}}{\eta} \|se_\delta\|_{\mathcal{H}} + |s| \frac{C_{\ell,\delta}}{\eta} \|G_1 e_\delta\|_{\mathcal{G}} + \frac{|s|}{\eta} C_p (c_\ell + C_b \|su_0\|_{D(G_1)}) \|G_0 z_\delta\|_{\mathcal{G}}. \quad \square \end{aligned} $$

Hence, $\mathcal{E}_\delta \le C_\delta \mathcal{E}_\delta^{\frac{1}{2}}$ which concludes the proof.

**4.2. Strong scale separation.** For conciesness, the following notation is introduced

$$ \mathcal{V}_\infty = D(G_0) \cap D(G_1). $$

We introduce the bounded operator $\mathbf{G}_0 : \mathcal{V}_\infty \to \mathcal{G}$ defined by

$$ \forall v \in \mathcal{V}_\infty, \quad \mathbf{G}_0 v = G_0 v. $$

The main theorem of this section relies on the assumption that the range of $\mathbf{G}_0$ is closed. It is equivalent to an *inf-sup condition* (see [5] for more details).

**ASSUMPTION 4.2.** The range of $\mathbf{G}_0$ is closed, namely there exits $C_c > 0$ such that

$$ (4.12) \qquad \forall u \in (\text{Ker } \mathbf{G}_0)^\perp \cap \mathcal{V}_\infty, \quad \|u\|_{\mathcal{V}_\infty} \le C_c \|\mathbf{G}_0 u\|_{\mathcal{G}}, $$

where the orthogonality holds for the scalar product in $\mathcal{V}_\infty$.
The adjoint of $G_0$ and $G_1$ are given by

$$ egin{cases} D(G_0^*) = \{U \in \mathcal{G} \mid \partial_y(\mu_0 U) \in \mathcal{H}, U(\cdot, 0) = U(\cdot, 1)\}, & G_0^* U = -\partial_y(\mu_0 U), \\ D(G_1^*) = \{U \in \mathcal{G} \mid \partial_x(\mu_0 U) \in \mathcal{H}, U(0, \cdot) = U(L, \cdot) = 0\}, & G_1^* U = -\partial_x(\mu_0 U). \end{cases} $$

One can check, using the density of smooth functions that the sequence of unbounded operators defined by

$$ D(G_{\delta}) = \mathcal{V}_{\delta} = H_{\delta}^{1, \delta}(\Omega \times Y), \quad G_{\delta} = \overline{\delta^{-1}G_{0} + G_{1}}, $$

is a sequence of two-scale operators. One can also check that problem (3.5) reads: for $f \in L^2(\Omega)$ given, find $U_{\delta}$ satisfying for all $V \in \mathcal{V}_{\delta}$,

$$ s^2(U_{\delta}, V)_{\mathcal{H}} + (G_{\delta} U_{\delta}, G_{\delta} V)_{\mathcal{G}} = (f, V)_{\mathcal{H}}. $$

**Limit problem.** The orthogonal projection operator into Ker $G_0^*$ denoted $Q_0$ has to be computed, one can check that it is given by

$$ Q_0 u = \frac{1}{\mu_0(x, y)} \left( \int_Y \frac{1}{\mu_0(x, y)} dy \right)^{-1} \left( \int_Y u(x, y) dy \right). $$

To write the limit problem one has to verify that $Q_0 G_1$ is closable. This can be done using Lemma 3.1 and verifying that

$$ \{\mu_0^{-1}\varphi \mid \varphi \in C_0^{\infty}(\Omega)\} \subset \text{Ker } G_0^* \cap D(G_1^*) $$

is dense in

$$ \text{Ker } G_0^* = \{u \in \mathcal{G} \mid \partial_y(\mu_0 u) = 0\}. $$

The limit problem is: for $f \in L^2(\Omega)$ given, find $U_0 \in \mathcal{V}_0^0$ with $V_0^0 := \text{Ker } G_0 \cap \mathcal{V}_0 = \text{Ker } G_0 \cap D(\overline{Q_0 G_1})$ such that for all $V \in \text{Ker } G_0 \cap D(G_1)$,

$$ s^2(U_0, V)_{\mathcal{H}} + (\overline{Q_0 G_1} U_0, \overline{Q_0 G_1} V)_{\mathcal{G}} = (f, v)_{\mathcal{H}}. $$

One can verify that

$$ D(\overline{Q_0 G_1}) = \{u \in \mathcal{H} \mid \partial_x \int_Y u(x, y) dy \in \mathcal{G}\}, $$

and for all $u \in D(\overline{Q_0 G_1})$

$$ \overline{Q_0 G_1} u = \frac{1}{\mu_0} \left( \int_Y \frac{1}{\mu_0(x, y)} dy \right)^{-1} \partial_x \int_Y u(x, y) dy. $$

Moreover the density result (2.15) holds trivially since, in that specific case,

$$ \text{Ker } G_0 \cap D(G_1) = \text{Ker } G_0 \cap D(\overline{Q_0 G_1}) = \mathcal{V}_0^0. $$

This implies that, we have strong convergence to the unique solution of the limit problem given by: find $U_0 \in \mathcal{V}_0^0$ such that for all $V \in \mathcal{V}_0^0$,

$$ s^2(U_0, V)_{\mathcal{H}} + (\overline{Q_0 G_1} U_0, \overline{Q_0 G_1} V)_{\mathcal{G}} = (f, V)_{\mathcal{H}}, $$
**THEOREM 4.3.** Let the strong scale separation property holds, Assumption 4.2 and let assume that there exists an anti-linear form $\ell_0 \in \mathcal{L}(D(G_1))$ such that

$$ \forall v \in \mathrm{Ker} G_0 \cap D(G_1), \quad |\ell_{\delta}(v) - \ell_0(v)| \le c_{\ell,\delta} \|v\|_{\mathcal{H}} + C_{\ell,\delta} \|G_1 v\|_{\mathcal{G}}. $$

Then, the sequence of solutions $\{u_\delta\} \subset D(G_0) \cap D(G_1)$ to (2.28), satisfies

$$ (4.13) \qquad (|||u_\delta - u_0|||^2 + \delta^{-2} \|G_0 u_\delta\|_{\mathcal{G}}^2)^{\frac{1}{2}} \le C_\delta $$

where

$$ C_{\delta} = \frac{1}{\eta} \max(c_{\ell,\delta}, |s|C_{\ell,\delta}, \delta \|sg_0\|_{\mathcal{G}}), $$

and $g_0$ is the Lagrange multiplier associated to the mixed form of the limit problem: find $(u_0, g_0) \in \mathcal{V}_\infty \times \mathrm{Im} \, \mathbf{G}_0$ such that for all, $v \in \mathcal{V}_\infty$,

$$ (4.14) \qquad \begin{cases} s^2(u_0, v)_{\mathcal{H}} + (G_1 u_0, G_1 v)_{\mathcal{G}} + b(su_\delta, v) + (g_0, G_0 v)_{\mathcal{G}} = \ell_0(v), \\ G_0 u_0 = 0. \end{cases} $$

Proof.

**Step 1: Introduction of a Lagrange multiplier.** For all $v \in \mathcal{V}_\infty$ a functional $L \in \mathcal{L}(\mathcal{V}_\infty)$ is defined by

$$ (4.15) \qquad L(v) = \ell_0(v) - s^2(u_0, v)_{\mathcal{H}} - (G_1 u_0, G_1 v)_{\mathcal{G}} - b(su_0, v). $$

From the Riesz representation theorem there exists a unique element $\ell \in \mathcal{V}_\infty$ such that

$$ \forall v \in \mathcal{V}_\infty, \quad (\ell, v)_{\mathcal{V}_\infty} = L(v). $$

Observe now that, since (2.29) holds, $(\ell, v)_{\mathcal{V}_\infty} = 0$ for all

$$ v \in \mathrm{Ker} G_0 \cap D(G_1) = \mathrm{Ker} \, \mathbf{G}_0 \cap D(G_1). $$

Hence, $\ell \in (\mathrm{Ker} \, \mathbf{G}_0)^\perp$ (here the orthogonality relation holds for the scalar product in $\mathcal{V}_\infty$). Now, observe that

$$ (\mathrm{Ker} \, \mathbf{G}_0)^\perp = \overline{\mathrm{Im} \, \mathbf{G}_0^*}. $$

Since the range of $\mathbf{G}_0$ is closed by Assumption 4.2, the range of $\mathbf{G}_0^*$ is closed, hence $\ell$ belongs to the range of $\mathbf{G}_0^*$. More precisely,

$$ \ell \in \mathrm{Im} \, \mathbf{G}_0^* \quad \Rightarrow \quad \exists g \in D(\mathbf{G}_0^*) \text{ such that } \ell = \mathbf{G}_0^* g. $$

The result above can be slightly improved. Since $\mathbf{G}_0^*$ has closed range, there exists $C_0 > 0$ such that

$$ \forall g \in (\mathrm{Ker} \, \mathbf{G}_0^*)^\perp, \quad \|g\|_{\mathcal{G}} \le C_0 \|\mathbf{G}_0^* g\|_{\mathcal{V}_\infty}, $$

showing that $\mathbf{G}_0^*$ is injective on $(\mathrm{Ker} \, \mathbf{G}_0^*)^\perp = \overline{\mathrm{Im} \, \mathbf{G}_0} = \mathrm{Im} \, \mathbf{G}_0$. Hence the following unique characterization of $\ell$ is deduced,

$$ \ell \in \mathrm{Im} \, \mathbf{G}_0^* \quad \Rightarrow \quad \exists! g_0 \in \mathrm{Im} \, \mathbf{G}_0 \text{ such that } \ell = \mathbf{G}_0^* g_0. $$
where

$$ (\overline{Q_0 G_1 U_0, \overline{Q_0 G_1 V}})_G = \int_{\Omega} \left( \int_{Y} \frac{1}{\mu_0(x, y)} dy \right)^{-1} \partial_x U_0(x) \partial_x V(x) dx $$

and

$$ (U_0, V)_H = \int_{\Omega} U_0(x) V(x) dx. $$

The limit problem is therefore a wave propagation problem with velocity the square root of the geometric mean of $\mu_0$. This is exactly the well-known result that we were expecting. Remark however that the obtained strong convergence result is the following

$$ \|U_{\delta} - U_{0}\|_{H}^{2} + \|\overline{Q_{0} G_{1}}(U_{\delta} - U_{0})\|_{G}^{2} \underset{\delta \to 0}{\longrightarrow} 0, $$

which implies in particular (below $P_0$ is the orthogonal projection operator into the kernel of $G_0$),

$$ \|(I - P_0)U_{\delta}\|_{H}^{2} + \|P_0(U_{\delta} - U_0)\|_{H}^{2} \underset{\delta \to 0}{\longrightarrow} 0. $$

These convergence results correspond to

$$ \int_{\Omega} \left| \left( \int_{Y} U_{\delta}(x, y) dy \right) - U_{0}(x) \right|^{2} dx + \int_{\Omega} \left| \partial_{x} \left( \int_{Y} U_{\delta}(x, y) dy \right) - \partial_{x} U_{0}(x) \right|^{2} dx \underset{\delta \to 0}{\longrightarrow} 0 $$

and

$$ \int_{\Omega \times Y} \left( U_{\delta}(x, y) - \int_{Y} U_{\delta}(x, z) dz \right)^{2} dx dy \underset{\delta \to 0}{\longrightarrow} 0. $$

Coming back to the original function $u_{\delta,z}$ this shows that

$$ \int_{\Omega} |\langle u_{\delta,z} \rangle - U_0(x)|^2 dx + \int_{\Omega} |\partial_x \langle u_{\delta,z} \rangle - \partial_x U_0(x)|^2 dx \underset{\delta \to 0}{\longrightarrow} 0, $$

and

$$ \int_{\Omega \times Y} (u_{\delta,z} - \langle u_{\delta,z} \rangle)^2 dx dz \underset{\delta \to 0}{\longrightarrow} 0, $$

where $\langle u_{\delta,z} \rangle$ is the average over $z$ of $u_{\delta,z}$. These convergence results shows, first, that $u_{\delta,z}$ converges strongly to its average, second, that the average $\langle u_{\delta,z} \rangle$ strongly converges in $H^1(\Omega)$ towards $U_0$. These convergence results are obtained with only (3.2) as requirement on the regularity of $\mu_0$. Finally, note that, in order to continue the analysis one should relate $u_{\delta}$ to $U_{\delta}$ (or $u_{\delta,z}$) and $U_0$. Such relation, is studied in [30] and leads to convergence estimates.

**Remark 3.2.** Note, that the limit problem is obtained without having to define a cell problem as commonly done in the homogenization theory. The cell problem is in fact hidden in the definition of the projection operator $Q_0$. Indeed, one way to compute $Q_0g$ is to solve the problem

$$ (3.6) \qquad Q_0 g = g - G_0 v \quad \text{with} \quad G_0^* G_0 v = G_0^* g, $$

where $v \in \text{Ker} G_0^\perp$. The standard cell problem is recovered by choosing $g = 1$, indeed, $v$ solution to (3.6) is then the 1-periodic solution to $-\partial_y(\mu_0 \partial_y v) = -\partial_y \mu_0$.
Using (4.15) and the property of the adjoint

$$ \forall v \in \mathcal{V}_{\infty}, \quad L(v) = (\ell, v)_{\mathcal{V}_{\infty}} = (\mathbf{G}_{0}^{*} g_{0}, v)_{\mathcal{V}_{\infty}} = (g_{0}, \mathbf{G}_{0} v)_{\mathcal{G}} = (g_{0}, G_{0} v)_{\mathcal{G}}, $$

it has been shown that there exists a unique $g_0 \in \operatorname{Im} \mathbf{G}_0$ such that (4.14) is satisfied.

**Step 2: Mixed formulation for the family of problems.** The objective is now to write a formulation for the family of problems that is similar to the one obtained just above. An element $g_{\delta} \in \operatorname{Im} \mathbf{G}_0$ is introduced,

$$ g_{\delta} = \delta^{-2} G_{0} u_{\delta} \in \operatorname{Im} \mathbf{G}_{0}. $$

Problem (2.28) is equivalent to: for all $v \in \mathcal{V}_{\delta}$,

$$ (4.16) \qquad \begin{cases} s^2(u_{\delta}, v)_{\mathcal{H}} + (G_1 u_{\delta}, G_1 v)_{\mathcal{G}} + b(s u_{\delta}, v) + (g_{\delta}, G_0 v)_{\mathcal{G}} = \ell_{\delta}(v), \\ G_0 u_{\delta} = \delta^2 g_{\delta}. \end{cases} $$

**Step 3: Energy identity for the error equation.** The following error terms are introduced,

$$ e_{\delta} = u_{\delta} - u_{0} \in \mathcal{V}_{\infty}, \quad h_{\delta} = g_{\delta} - g_{0} \in \operatorname{Im} \mathbf{G}_{0}. $$

Using (4.14) and (4.16) one finds that: for all $v \in \mathcal{V}_{\infty}$,

$$ (4.17) \qquad \begin{cases} s^2(e_{\delta}, v)_{\mathcal{H}} + (G_1 e_{\delta}, G_1 v)_{\mathcal{G}} + b(se_{\delta}, v)_{\mathcal{G}} + (h_{\delta}, G_0 v)_{\mathcal{G}} \\ = \ell_{\delta}(v) - \ell_0(v), \\ G_0 e_{\delta} = \delta^2 h_{\delta} + \delta^2 g_0. \end{cases} $$

Choosing $v = s e_{\delta}$, it comes

$$ (4.18) \quad s\|se_{\delta}\|_{\mathcal{H}} + \bar{s}(G_1 e_{\delta}, G_1 e_{\delta})_{\mathcal{G}} + b(se_{\delta}, se_{\delta}) + \bar{s}(h_{\delta}, G_0 e_{\delta})_{\mathcal{G}} = \ell_{\delta}(se_{\delta}) - \ell_0(se_{\delta}). $$

The second equation of (4.17) can be used to simplify the identity just above, indeed,

$$ (4.19) \qquad (h_{\delta}, G_0 e_{\delta})_{\mathcal{G}} = \delta^2 \|h_{\delta}\|_{\mathcal{G}}^2 + \delta^2 (h_{\delta}, g_0)_{\mathcal{G}}. $$

Using (4.19) in (4.18) the following energy identity can be derived,

$$ (4.20) \qquad \|\|e_{\delta}\|\|^2 + \delta^2 \|h_{\delta}\|_{\mathcal{G}}^2 = \frac{1}{\eta} \Re(e(\ell_{\delta}(se_{\delta}) - \ell_0(se_{\delta}) - \delta^2(h_{\delta}, sg_0)_{\mathcal{G}}), $$

leading to the estimation

$$ \|\|e_{\delta}\|\|^2 + \delta^2 \|h_{\delta}\|_{\mathcal{G}}^2 \le \frac{1}{\eta} (c_{\ell, \delta} \|se_{\delta}\|_{\mathcal{H}} + |s| C_{\ell, \delta} \|G_1 e_{\delta}\|_{\mathcal{G}}) + \frac{\delta}{\eta} \|sg_0\|_{\mathcal{G}} \delta \|h_{\delta}\|_{\mathcal{G}}. $$

Finally, one can deduce that $\|\|e_{\delta}\|\|^2 + \delta^2 \|h_{\delta}\|_{\mathcal{G}}^2 \le C_{\delta}^2$ with $C_{\delta}$ as defined in the statement of the Theorem. □
**4. Applications in elastodynamics and piezoelectricity.** For illustration, two examples in elastic wave propagation are discussed as well as a problem in piezoelectricity. In this section, $\Omega \subset \mathbb{R}^3$ is a simply connected Lipschitz bounded domain and its boundary has a single component. We denote by $\Sigma$ the space of symmetric second-order tensor. The space $L_C^2(\Omega; \Sigma)$ is the space of square integrable second order symmetric tensor field equipped with $(\cdot, \cdot)_{L_C^2(\Omega; \Sigma)} = (C \cdot, \cdot)_{L^2(\Omega)^{3 \times 3}}$. where $C$ is uniformly coercive elasticity tensor field, $u$ is a vector field, $\varepsilon(u)$ its symmetrized gradient. The density $\rho$ is a scalar field uniformly bounded by above and below.

**4.1. Nearly-incompressible elastodynamics.**

**Nondimensionalized problem.** After nondimensionalization the problem of wave propagation in nearly-incompressible media reads: for $f \in L^2(\Omega)^3$ given, find $u_\delta \in H_0^1(\Omega)^3$ such that for all $v \in H_0^1(\Omega)^3$,

$$ (4.1) \quad s^2 \int_{\Omega} \rho u_{\delta} \cdot \bar{v} \, dx + \delta^{-2} \int_{\Omega} \operatorname{div} u_{\delta} \operatorname{div} \bar{v} \, dx + \int_{\Omega} C \varepsilon(u_{\delta}) : \varepsilon(\bar{v}) \, dx = \int_{\Omega} \rho f \cdot \bar{v} \, dx. $$

Defining $\mathcal{H} = L_{\rho}^2(\Omega)^3$ the space of functions in $L^2(\Omega)^3$ equipped with the scalar product $(\cdot, \cdot)_\mathcal{H} = (\rho \cdot, \cdot)_{L^2(\Omega)^3}$ and $\mathcal{G} = L^2(\Omega) \times L_C^2(\Omega; \Sigma)$ the problem at hand can be rewritten: for $f \in L^2(\Omega)^3$ given, find $u_\delta \in H_0^1(\Omega)^3$ such that

$$ \forall v \in H_0^1(\Omega)^3, \quad s^2(u_\delta, v)_\mathcal{H} + \left( \begin{pmatrix} \delta^{-1} \operatorname{div} u_\delta \\ \varepsilon(u_\delta) \end{pmatrix}, \begin{pmatrix} \delta^{-1} \operatorname{div} v \\ \varepsilon(v) \end{pmatrix} \right)_\mathcal{G} = (f, v)_\mathcal{H}. $$

The formulation above can be easily recast in the form (2.6) using the sequence of two-scale operators defined below.

**The sequence of two-scale operators.** Closed and densely defined operators $G_0$ and $G_1$, from $\mathcal{H}$ to $\mathcal{G}$ are defined by

$$ D(G_0) = H_0(\operatorname{div}, \Omega), \quad G_0 u = \begin{pmatrix} \operatorname{div} u \\ 0 \end{pmatrix}, \quad D(G_1) = H_0^1(\Omega)^3, \quad G_1 u = \begin{pmatrix} 0 \\ \varepsilon(u) \end{pmatrix}, $$

where $H_0(\operatorname{div}, \Omega)$ is the space of function with square integrable divergence and vanishing normal trace. The sequence of operators $G_\delta = \delta^{-1}G_0 + G_1$ is a sequence of formal two-scale operators. It is easy to verify that $\operatorname{Im} G_0 \perp \operatorname{Im} G_1$. Moreover, from [10] we have that

$$ D(G_0) \cap D(G_1) = H_0^1(\Omega)^3 \text{ is dense in } D(G_0) = H_0(\operatorname{div}, \Omega). $$

Thus, the strong scale property holds (hence in particular $\mathcal{V}_0 = D(G_1)$). The solution to (4.1) converges strongly in $H_0^1(\Omega)^3$ to the solution of the limit problem given below.

**Limit problem.** Setting

$$ \mathcal{V}_0^0 = \operatorname{Ker} G_0 \cap \mathcal{V}_0 = \operatorname{Ker} G_0 \cap D(G_1) = \{u \in H_0^1(\Omega)^3 | \operatorname{div} u = 0\}, $$

the limit problem is given by: for $f \in L^2(\Omega)^3$ given, find $u_0 \in \mathcal{V}_0^0$ such that

$$ \forall v \in \mathcal{V}_0^0, \quad s^2(u_0, v) + (G_1 u_0, G_1 v)_\mathcal{G} = s^2(u_0, v) + (\varepsilon(u_0), \varepsilon(v))_{L^2(\Omega; C)} = (f, v)_\mathcal{H}. $$

This is the standard incompressible limit (see [28] Section 58). Note that

$$ \mathcal{R}_0 = \operatorname{Ker} G_1 \cap \operatorname{Ker} G_0 = \{u \in H_0^1(\Omega)^3 | \varepsilon(u) = 0\} = \{0\}, $$

namely, the space of quasi-static solution $\mathcal{R}_0$ is reduced to $\{0\}$.
**5. Proofs of Section 2.** The proofs below use several results in operator theory that the reader can find in [11] Chap. 12, or [15] Chap 3. Section 5.

**Proof of Theorem 2.3.** Only (2.4)(a) is proven, the property (2.4)(b) is proven similarly. Since $u_\delta$ is uniformly bounded in $\mathcal{H}$ it converges, up to a subsequence, weakly to $u_0$ in $\mathcal{H}$, for all $h \in D(G_0^*) \cap D(G_1^*)$ their holds,

$$ (u_\delta, G_0^* h)_\mathcal{H} \underset{\delta \to 0}{\longrightarrow} (u_0, G_0^* h)_\mathcal{H}. $$

When $(u_\delta, G_0^* h)_G \underset{\delta \to 0}{\longrightarrow} 0$ it holds

$$ (u_0, G_0^* h)_\mathcal{H} = 0, \quad \forall h \in D(G_0^*) \cap D(G_1^*). $$

By density of $D(G_0^*) \cap D(G_1^*)$ into $D(G_0^*)$ for the graph norm of $G_0^*$ it can be deduced that the equality above holds for all $h \in D(G_0^*)$. Since $G_0$ is closed and densely defined, this shows that $u_0 \in D(G_0)$ and $G_0 u_0 = 0$.

**Proof of Theorem 2.5.**

**The operator $G_1^* Q_0$ is densely defined.** By definition

$$ D(G_1^* Q_0) = \{g \in D(Q_0) = \mathcal{G} \mid Q_0 g \in D(G_1^*)\} $$

For all $g \in \mathcal{G}$, an orthogonal decomposition in $\mathcal{G}$ holds. It is given by $g = g_0 + g_\perp$ with $g_0 \in \text{Ker } G_0^*$ and $Q_0 g_\perp = 0$. Therefore

$$ D(G_1^* Q_0) = \{g = g_0 + g_\perp, g_0 \in \text{Ker } G_0^*, g_\perp \in (\text{Ker } G_0^*)^\perp \mid Q_0 g = g_0 \in D(G_1^*)\}, $$

which shows that

$$ (5.1) \qquad D(G_1^* Q_0) = (\text{Ker } G_0^* \cap D(G_1^*)) \oplus (\text{Ker } G_0^*)^\perp. $$

By assumption $\text{Ker } G_0^* \cap D(G_1^*)$ is dense in $\text{Ker } G_0^*$ and since

$$ \mathcal{G} = \text{Ker } G_0^* \oplus (\text{Ker } G_0^*)^\perp, $$

it ensues that $D(G_1^* Q_0)$ is dense in $\mathcal{G}$.

**The operator $Q_0 G_1$ is closable.** Since it has been shown that $G_1^* Q_0$ is densely defined, it has an adjoint. Since $Q_0$ is self-adjoint the following relation holds,

$$ Q_0 G_1 = Q_0^* G_1^{**} \subset (G_1^* Q_0)^*, $$

meaning that $(G_1^* Q_0)^*$ is a (closed) extension of $Q_0 G_1$. This last property implies that $Q_0 G_1$ is closable.

**Closure of $Q_0 G_\delta$.** Thanks to (2.3), the operator $\delta^{-1} G_0 + G_1$ is densely defined, its adjoint satisfies

$$ G_\delta^* = (\delta^{-1} G_0 + G_1)^* = \overline{\delta^{-1} G_0 + G_1}^* \supset (\delta^{-1} G_0^* + G_1^*) $$
## 4.2. Elastodynamics in plates.

**Nondimensionalized problem.** The problem of elastic wave propagation in a thin plate is now investigated. The thickness of the plate – in the $x_3$-direction – is assumed to be equal to $2\delta$. After nondimensionalization, the elastodynamics equations are written in a fixed domain $\Omega = S \times [-1, 1]$ where $S \subset \mathbb{R}^2$ is a Lipschitz bounded domain. For simplicity, we consider that the elastic tensor field is isotropic,

$$ \forall \varepsilon \in \Sigma, \quad \mathbf{C}\varepsilon = \lambda \sum_{j=1}^{3} \varepsilon_{jj} I + 2\mu\varepsilon, $$

where $I$ is the identity tensor in $\Sigma$ and where $\lambda$ and $\mu$ are the space dependent Lamé parameters of the medium. The problem to solve reads: for $f \in L^2(\Omega)^3$ given, find $u_\delta \in H^1(\Omega)^3$ such that, for all $v \in H^1(\Omega)^3$,

$$ s^2 \int_{\Omega} \rho u_{\delta} \cdot \bar{v} \, dx + \int_{\Omega} \mathbf{C} (\delta^{-1}\varepsilon_3(u) + \varepsilon_{\sigma}(u)) : (\delta^{-1}\varepsilon_3(\bar{v}) + \varepsilon_{\sigma}(\bar{v})) \, dx = \int_{\Omega} \rho f \cdot \bar{v} \, dx. $$

where

$$ \varepsilon_{\sigma}(u) = \frac{1}{2} \begin{pmatrix} 2\partial_1 u_1 & \partial_1 u_2 + \partial_2 u_1 & \partial_1 u_3 \\ \partial_1 u_2 + \partial_2 u_1 & 2\partial_2 u_2 & \partial_2 u_3 \\ \partial_1 u_3 & \partial_2 u_3 & 0 \end{pmatrix}, $$

and

$$ \varepsilon_3(u) = \frac{1}{2} \begin{pmatrix} 0 & 0 & \partial_3 u_1 \\ 0 & 0 & \partial_3 u_2 \\ \partial_3 u_1 & \partial_3 u_2 & 2\partial_3 u_3 \end{pmatrix}. $$

Note that $\varepsilon(u) = \varepsilon_3(u) + \varepsilon_{\sigma}(u)$. To write these equations in the abstract setting, the space $\mathcal{H}$ is defined by $\mathcal{H} = L^2(\Omega)^3$ and $\mathcal{G}$ by $\mathcal{G} = L^2_{\mathbb{C}}(\Omega; \Sigma)$.

**The sequence of two-scale operators.** Closed and densely defined operators $G_0$ and $G_1$, from $\mathcal{H}$ to $\mathcal{G}$ are defined by

$$ D(G_0) = \{u \in L^2(\Omega)^3 \mid \varepsilon_3(u) \in L^2(\Omega; \Sigma)\} \quad G_0 u = \varepsilon_3(u), $$

and

$$ D(G_1) = \{u \in L^2(\Omega)^3 \mid \varepsilon_{\sigma}(u) \in L^2(\Omega; \Sigma)\} \quad G_1 u = \varepsilon_{\sigma}(u), $$

One can show that $\delta^{-1}G_0 + G_1$ is densely defined, and closed, i.e.,

$$ G_{\delta} = \overline{\delta^{-1}G_{0} + G_{1}} = \delta^{-1}G_{0} + G_{1}, $$

moreover,

$$ D(G_{\delta}) = \{u \in L^2(\Omega)^3 \mid \varepsilon(u) \in L^2(\Omega; \Sigma)\} = H^1(\Omega)^3, \quad G_{\delta}u = \delta^{-1}\varepsilon_3(u) + \varepsilon_{\sigma}(u). $$

Note that the Korn's inequality has been used to deduce that $D(G_\delta) = H^1(\Omega)^3$. The problem now reads: for $f \in L^2(\Omega)^3$ given, find $u_\delta \in H^1(\Omega)^3$ such that

$$ \forall v \in H^1(\Omega)^3, \quad s^2(u, v)_{\mathcal{H}} + (G_\delta u, G_\delta v)_{\mathcal{G}} = (f, v)_{\mathcal{H}}. $$

To check that $G_\delta$ is a sequence of two-scale operators the adjoint $G_0$ and $G_1$ have to be computed. In order to define these adjoints, the following divergence operators are introduced

$$ \forall \varepsilon \in L^2(\Omega; \Sigma), \quad \mathbf{Div}_3 = \sum_{j=1}^{3} \partial_3 \varepsilon_{j3} \mathbf{e}_j, \quad \mathbf{Div}_{\sigma} = \mathbf{Div} - \mathbf{Div}_3, $$
Therefore,

$$ (5.2) \qquad G_1^* Q_0 = (\delta^{-1} G_0^* + G_1^*) Q_0 \subset G_\delta^* Q_0 = (Q_0 G_\delta)^*. $$

It can be deduced, first, from (5.2) that $(Q_0 G_\delta)^*$ is also densely defined and therefore $Q_0 G_\delta$ is closable; second, from (5.2), and the property that $Q_0$ is bounded and self-adjoint, that

$$ \overline{Q_0 G_\delta} = (Q_0 G_\delta)^{**} \subset (G_1^* Q_0)^* = (Q_0 G_1)^{**} = \overline{Q_0 G_1}. $$

**Proof of Theorem 2.10.** Our objective is to prove that

$$ \left\{ \begin{array}{l} Q_0 G_1 = G_1 \\ \text{Ker } G_0^* \cap D(G_1^*) \text{ dense in } \text{Ker } G_0^*, \\ D(G_0^*) \cap D(G_1^*) \text{ dense in } D(G_0^*), \\ \delta^{-1} G_0 + G_1 \text{ is closed.} \end{array} \right. $$

The other statements of the theorem are direct consequences of the above properties.

**The equality $Q_0 G_1 = G_1$.** By assumption,

$$ \forall (u, v) \in D(G_0) \times D(G_1) \quad (G_0 u, G_1 v)_G = 0. $$

By density, this, implies

$$ \forall (g, v) \in \overline{\text{Im}(G_0)} \times D(G_1) \quad (g, G_1 v)_G = 0, $$

where $\overline{\text{Im}(G_0)}$ is the closure of the range of $G_0$ for the norm in $\mathcal{G}$. Now, since $I - Q_0$ is, by definition, the orthogonal projection operator on the closure of $\text{Im } G_0$, there holds

$$ \forall (g, v) \in \mathcal{G} \times D(G_1) \quad ((I - Q_0)g, G_1 v)_G = 0, $$

also meaning that, for all $v \in D(G_1)$, $G_1 v = Q_0 G_1 v$.

**Density results.** It has been shown that $Q_0 G_1 = G_1$ hence $Q_0 G_1$ is closed and densely defined. Therefore $(Q_0 G_1)^* = G_1^* G_0^*$ is closed and densely defined in $\mathcal{G}$. More precisely,

$$ D(G_1^* Q_0) = (\text{Ker } G_0^* \cap D(G_1^*)) \oplus (\text{Ker } G_0^*)^\perp \text{ dense in } \mathcal{G} = \text{Ker } G_0^* \oplus (\text{Ker } G_0^*)^\perp. $$

From the statement it can be deduced that $\text{Ker } G_0^* \cap D(G_1^*)$ is dense in $\text{Ker } G_0^*$. It must be checked now that $D(G_0^*) \cap D(G_1^*)$ is dense in $D(G_0^*)$ for the graph norm of $G_0^*$. The following vectorial space is defined,

$$ \mathcal{Z} = (\text{Ker } G_0^* \cap D(G_1^*)) \oplus ((\text{Ker } G_0^*)^\perp \cap D(G_0^*)), $$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{G}$. Observe that $\mathcal{Z}$ is a subspace of $D(G_0^*) \cap D(G_1^*)$ since by assumption

$$ (\text{Ker } G_0^*)^\perp = \overline{\text{Im } G_0} \subset \text{Ker } G_1^*. $$
where $e_j$ is the jth element of the canonical basis and $\mathbf{Div}$ is the standard divergence operator of a second order tensor field. The boundary of $\Omega$ is decomposed using the following two disjoints subdomains

$$ \Gamma_S = S \times \{-1, 1\}, \quad \Gamma_\sigma = \partial S \times (-1, -1). $$

The unitary outward normal on $\Gamma_\sigma$ is denoted $\mathbf{n}_\sigma$. With these notations, the adjoints are given by

$$ \left\{ \begin{array}{l} D(G_0^*) = \{\varepsilon \in L^2(\Omega; \Sigma) \mid \mathbf{Div}_3(\mathbf{C}\varepsilon) \in L^2(\Omega)^3, (\mathbf{C}\varepsilon)\mathbf{e}_3 = 0 \text{ on } \Gamma_S\}, \\ D(G_1^*) = \{\varepsilon \in L^2(\Omega; \Sigma) \mid \mathbf{Div}_\sigma(\mathbf{C}\varepsilon) \in L^2(\Omega)^3, (\mathbf{C}\varepsilon)\mathbf{n}_\sigma = 0 \text{ on } \Gamma_\sigma\}, \end{array} \right. $$

and

$$ G_0^*\varepsilon = -\rho^{-1}\mathbf{Div}_3(\mathbf{C}\varepsilon), \quad G_1^*\varepsilon = -\rho^{-1}\mathbf{Div}_\sigma(\mathbf{C}\varepsilon). $$

It can be verified that $D(G_0) \cap D(G_1)$ is dense in $D(G_0)$ and $D(G_0^*) \cap D(G_1^*)$ is dense in $D(G_0^*)$. The proof simply amounts to use the density of smooth functions in $H^1$-spaces.

**Limit problem.** To deduce the limit problem the projection operator on Ker $G_0^*$ has to be computed. Here

$$ \mathrm{Ker} \, G_0^* = \{ \varepsilon \in L^2(\Omega; \Sigma) \mid (\mathbf{C}\varepsilon)\mathbf{e}_3 = 0 \text{ in } \Omega \}, $$

and for isotropic media (the isotropy is used here for simplicity) one can check that the orthogonal projection operator on Ker $G_0^*$ is given by

$$ \forall \varepsilon \in L_C^2(\Omega; \Sigma), \quad Q_0\varepsilon = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & 0 \\ \varepsilon_{12} & \varepsilon_{22} & 0 \\ 0 & 0 & \frac{-\lambda(\varepsilon_{11} + \varepsilon_{22})}{\lambda + 2\mu} \end{pmatrix}. $$

Note that one recovers the well-known in-plane strain tensor corresponding to plane-stress problems. See for instance [28] Section 45 for more details. After a straightforward formal computation of $Q_0G_1u$, one can then show that, setting

$$ \mathcal{V}_0^0 = \mathrm{Ker} \, G_0 \cap D(\overline{Q_0G_1}) = \{u \in L^2(\Omega)^3 \mid \partial_3 u = 0, u_1 \in H^1(\Omega), u_2 \in H^1(\Omega)\}, $$

the limit problem is given by: for $f \in \mathcal{H}$ given, find $u_0 \in \mathcal{V}_0^0$ such that

$$ (4.2) \quad \forall v \in \mathcal{V}_0^0, \quad s^2(u_0, v) + (\alpha\lambda\mathrm{div}_\sigma u_0, \mathrm{div}_\sigma v) + 2(\mu\varepsilon_\sigma(u_0), \varepsilon_\sigma(v))_{L^2(\Omega;\Sigma)} = (f, v)_\mathcal{H}, $$

where $\mathrm{div}_\sigma v = \partial_1 v_1 + \partial_2 v_2$ and $\alpha = 2\mu/(\lambda + 2\mu)$. It should be noted here that $u_3$ is not sought in $H^1(\Omega)$. Even more remarkable is the fact that the space of quasi-static solution

$$ \begin{array}{l} (4.3) \quad \mathcal{R}_0 := \mathrm{Ker} \, G_0 \cap \mathrm{Ker} \, \overline{Q_0G_1} \\ \qquad = \{u \in L^2(\Omega)^3 \mid \partial_3 u = 0, \partial_1 u_1 = \partial_2 u_2 = \partial_1 u_2 + \partial_2 u_1 = 0 \text{ in } \Omega\}, \end{array} $$

is of infinite dimension: it includes plane rigid-body motions and all square integrable displacements that are polarized only in the $x_3$-direction and independent of $x_3$.
Observe, moreover, that

$$D(G_0^*) = \mathrm{Ker} \, G_0^* \oplus ((\mathrm{Ker} \, G_0^*)^{\perp} \cap D(G_0^*))$$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{G}$. However, since it has been shown that $\mathrm{Ker} \, G_0^* \cap D(G_1^*)$ is dense in $\mathrm{Ker} \, G_0^*$ for the norm $\mathcal{H}$, it is therefore dense for the graph norm of $G_0^*$. This shows that $\mathcal{Z}$ (hence $D(G_0^*) \cap D(G_1^*)$) is dense in $D(G_0^*)$ for the same norm.

**The operator $\delta^{-1}G_0 + G_1$ is closed.** The graph of the operator $\delta^{-1}G_0 + G_1$ is given by

$$\mathcal{G}_r(\delta^{-1}G_0 + G_1) = \{(u, g) \in \mathcal{H} \times \mathcal{G} \mid u \in D(G_0) \cap D(G_1), g = \delta^{-1}G_0u + G_1u\}.$$

It must be verified that the graph is closed for the composite norm in $\mathcal{H} \times \mathcal{G}$. Let $\{u_n\}$ be a sequence in $D(G_0) \cap D(G_1)$ such that, for some $(u, g) \in \mathcal{H} \times \mathcal{G}$,

$$u_n \to_{\mathcal{H}} u, \quad \delta^{-1}G_0u_n + G_1u_n \to_{\mathcal{G}} g.$$

This implies

$$Q_0G_1u_n = G_1u_n \to_{\mathcal{G}} Q_0g.$$

However, since $G_1$ is closed, $u \in D(G_1)$ and $Q_0g = G_1u$. Consequently

$$u_n \to_{\mathcal{H}} u, \quad \delta^{-1}G_0u_n \to_{\mathcal{G}} (I - Q_0)g,$$

and since $G_0$ is closed, $u \in D(G_0)$ and $(I - Q_0)g = \delta^{-1}G_0u$ which conclude the proof since

$$u \in D(G_0) \cap D(G_1) \text{ and } g = \delta^{-1}G_0u + G_1u.$$

**Proof of Theorem 2.12.** Our objective is to prove that

$$\begin{cases} G_1P_0 = G_1, \\ D(G_0) \cap D(G_1) \text{ dense in } D(G_0), \end{cases}$$

using that, by assumption,

$$\text{(5.3)} \qquad \left\{ \begin{array}{ll} (\mathrm{Ker} \, G_1)^{\perp} = \overline{\mathrm{Im}(G_1^*)} \subset \mathrm{Ker} \, G_0, & (a) \\ (\mathrm{Ker} \, G_0)^{\perp} = \overline{\mathrm{Im}(G_0^*)} \subset \mathrm{Ker} \, G_1. & (b) \end{array} \right. $$

As a direct consequence of (5.3)(a) it can be deduced that

$$P_0G_1^* = G_1^* \Rightarrow G_1P_0 = G_1.$$

Therefore $D(G_1P_0) = \mathrm{Ker} \, G_0 \cap D(G_1) \oplus (\mathrm{Ker} \, G_0)^{\perp}$ is dense in $\mathcal{G}$, which implies that

$$\text{(5.4)} \qquad \mathrm{Ker} \, G_0 \cap D(G_1) \text{ dense in } \mathrm{Ker} \, G_0.$$

We define the space

$$\mathcal{W} = (\mathrm{Ker} \, G_0 \cap D(G_1)) \oplus ((\mathrm{Ker} \, G_0)^{\perp} \cap D(G_0)),$$
### 4.3. Piezoelectricity.

**Nondimensionalized problem.** Piezoelectric equations couple the electromagnetic field $(E, H)$ and the velocity field $v$ (where $v = \partial_t u$) thanks to a third order tensor field called the piezoelectric tensor and denoted $\mathbf{d}$. This tensor maps vector fields to tensor fields,

$$ \forall x \in \Omega, \quad \forall \mathbf{E} \in \mathbb{R}^3, \quad \mathbf{d}(x)\mathbf{E} \in \Sigma. $$

The objective of this section is to explain how the quasi-static assumption in piezoelectricity (in particular, the electric field reduces to a gradient of a potential) can be justified by the proposed abstract framework. After nondimensionalization, using Dirichlet boundary conditions on the boundary and assuming that the propagating media is surrounded by a perfectly conducting material we arrive at the following variational formulation (see [13] for more details): for $J \in L^2(\Omega)^3$ and $F \in L^2(\Omega)^3$ given, find $(E_\delta, v_\delta) \in H_0(\text{curl}; \Omega) \times H_0^1(\Omega)^3$ such that, for all $(\tilde{\mathbf{E}}, \tilde{\mathbf{v}}) \in H_0(\text{curl}; \Omega) \times H_0^1(\Omega)^3$,

$$ (4.4) \quad \left\{ \begin{array}{l} s^2(E_\delta, \tilde{E})_{L^2(\Omega)^3} + \delta^{-2}(\text{curl } E_\delta, \text{curl } \tilde{E})_{L^2(\Omega)^3} \\ \quad + s(\varepsilon(v_\delta), \mathbf{d}\tilde{E})_{L^2(\Omega)^3} = (J, \tilde{E})_{L^2(\Omega)^3}, \\[1.5ex] s^2(v_\delta, \tilde{v})_{L^2(\Omega)^3} + (\varepsilon(v_\delta), \varepsilon(\tilde{v}))_{L^2_G(\Omega; \Sigma)} \\ \quad - s(\varepsilon(\tilde{v}), \mathbf{d}E_\delta)_{L^2(\Omega)^3} = (F, \tilde{v})_{L^2(\Omega)^3}. \end{array} \right. $$

The small parameter $\delta$ represents the ratio of velocities of waves in solids and electromagnetic waves. The space $H_0(\text{curl}; \Omega)$ is the space of functions with square integrable curl and vanishing tangential trace.

**The sequence of two-scale operators.** By setting $\mathcal{H} = L^2(\Omega)^3 \times L^2(\Omega)^3$ and $\mathcal{G} = L^2(\Omega)^3 \times L^2_G(\Omega; \Sigma)$ and defining

$$ D(G_{\delta}) = \{ \Phi = \begin{pmatrix} \mathbf{E} \\ \mathbf{v} \end{pmatrix} \in H_0(\text{curl}; \Omega) \times H_0^1(\Omega)^3 \}, \quad G_{\delta}\Phi = G_{\delta} \begin{pmatrix} \mathbf{E} \\ \mathbf{v} \end{pmatrix} = \begin{pmatrix} \delta^{-1}\text{curl } \mathbf{E} \\ \varepsilon(\mathbf{v}) \end{pmatrix}, $$

problem (4.4) can be written as: for $\Psi = (J, F)^T \in \mathcal{H}$ given, find $\Phi_\delta \in D(G_\delta)$ such that, for all $\tilde{\Phi} \in D(G_\delta)$,

$$ (4.5) \quad s^2(\Phi_\delta, \tilde{\Phi})_{\mathcal{H}} + (G_\delta \Phi_\delta, G_\delta \tilde{\Phi})_{\mathcal{G}} + sb(\Phi_\delta, \tilde{\Phi}) = (\Psi, \tilde{\Phi})_{\mathcal{H}}, $$

with, for all $\Phi = \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_\delta)$ and $\tilde{\Phi} = \begin{pmatrix} \tilde{E} \\ \tilde{v} \end{pmatrix} \in D(G_\delta)$,

$$ b(\Phi, \tilde{\Phi}) = (\varepsilon(v), \mathbf{d}\tilde{E})_{L^2(\Omega)^3} - (\varepsilon(\tilde{v}), \mathbf{d}E)_{L^2(\Omega)^3}. $$

In order to show that (4.5) is a family of wave propagation problem as defined by (2.18) one needs to show first, that $G_\delta$ is a sequence of two-scale operators and second, that $b$ is a continuous sesquilinear form in $V_0 = D(\overline{Q_0G_1})$. The operators $G_0$ and $G_1$ are defined by

$$ D(G_0) = H_0(\text{curl}; \Omega) \times L^2(\Omega)^3, \quad \forall \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_0), \quad G_0 \begin{pmatrix} E \\ v \end{pmatrix} = \begin{pmatrix} \text{curl } E \\ 0 \end{pmatrix} $$

and

$$ D(G_1) = L^2(\Omega)^3 \times H_0^1(\Omega)^3, \quad \forall \begin{pmatrix} E \\ v \end{pmatrix} \in D(G_1) \quad G_1 \begin{pmatrix} E \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ \varepsilon(v) \end{pmatrix}. $$
where the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. Observe that the space $\mathcal{W}$ is a subspace of $D(G_0) \cap D(G_1)$ thanks to (5.3)(b). Moreover,

$$D(G_0) = \ker G_0 \oplus ((\ker G_0)^\perp \cap D(G_0)),$$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. It can be concluded that $\mathcal{W}$ is dense in $D(G_0)$ for the graph norm of $G_0$ using (5.4).

REFERENCES

[1] G Allaire. Shape optimization by the homogenization method. Springer, January 2002.

[2] A. Pazy (auth.). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer-Verlag New York, 1 edition, 1983.

[3] A Bensoussan and J.L. Lions. Asymptotic analysis for periodic structures. North Holland, 1978.

[4] A. Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter. Representation and Control of Infinite Dimensional Systems (Systems & Control: Foundations & Applications). Birkhauser, 2006.

[5] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. 1991.

[6] F Caforio and Sébastien Imperiale. Mathematical analysis of a penalization strategy for incompressible elastodynamics. Asymptotic Analysis, 2020.

[7] D. Chapelle and Bathe K.-J. The finite element analysis of shells - fundamentals. Computational Fluid and Solid Mechanics. Springer-Verlag Berlin Heidelberg, 2 edition, 2011.

[8] Philippe G. Ciarlet. Plates and junctions in elastic multi-structures : an asymptotic analysis. Recherches en matématiques appliquées 14. Masson, Springer-Verlag, 1990.

[9] R. Dautray and J.L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I. Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag Berlin Heidelberg, 1992.

[10] V. Girault and P.-A. Raviart. Finite element approximation of the Navier-Stokes equations. Technical report, Springer-Verlag, 1979.

[11] G. Grubb. Distributions and Operators. Graduate Texts in Mathematics. Springer New York, 2008.

[12] D. Huet. Phénomènes de perturbation singulière dans les problèmes aux limites. Annales de l'Institut Fourier, 10:61-150, 1960.

[13] S.Imperiale and P. Joly. Mathematical and numerical modelling of piezoelectric sensors. ESAIM: Mathematical Modelling and Numerical Analysis, 2012.

[14] S. Imperiale and P.Joly. Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section. Applied Numerical Mathematics, 2013.

[15] T. Kato. Perturbation Theory for Linear Operators. Classics in Mathematics. Springer Berlin Heidelberg, 1995.

[16] J.L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris, Dunod, 1969.

[17] J.L. Lions and E. Magenes. Problèmes aux limites non homogènes et Applications Vol. 1. Dunod, 1968.

[18] R. Temam. Navier-Stokes Equations: Theory and Numerical Analysis, Studies in mathematics and its application. North-Holland Publishing Company, 1979.

[19] M. Tucnak and G. Weiss. Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser Basel, 2009.

[20] G. Allaire. Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23 (6), pp.1482-1518. 1992.

[21] D. Cioranescu, A. Damlamian and G. Griso The Periodic Unfolding Method. Series in Contemporary Mathematics. Springer Nature 2018.

[22] A. Braides, Introduction to Homogenization and Gamma-Convergence, International Centre for Theoretical Physics. 1993.

[23] M. Waurick Homogenization of a class of linear partial differential equations Asymptotic Analysis, 82, 2013.

[24] B. Cockburn and P. Joly. Maxwell equations in polarizable media SIAM Journal on Mathematical Analysis, 19(6), 1988.

[25] G. Cohen, P. Joly, J. E. Roberts and N. Tordjman. High order triangular finite elements with mass lumping for the wave equation. SIAM Journal on Numerical Analysis, 38(6), 2001.
They are closed, densely defined operators. One can verify that the formal sequence of two-scale operators $G_{\delta} = \delta^{-1}G_{0} + G_{1}$ satisfies the strict scale separation property and in particular $\overline{Q_{0}G_{1}} = G_{1}$ and

$$ \mathcal{V}_{0} = D(\overline{Q_{0}G_{1}}) = D(G_{1}). $$

One can then verify easily that the sesquilinear form $b$ is indeed continuous in $\mathcal{V}_{0}$ if the tensor field $d$ is uniformly bounded in $\Omega$.

**Limit problem.** The limit problem reads: for $\Psi = (J, F)^{T} \in \mathcal{H}$ given, find $\Phi_{0} = (E_{0}, v_{0})^{T} \in \text{Ker } G_{0} \cap D(G_{1})$ such that, for all $\tilde{\Phi} = (\tilde{E}, \tilde{v})^{T} \in \text{Ker } G_{0} \cap D(G_{1})$,

$$ (4.6) \qquad s^{2}(\Phi_{0}, \tilde{\Phi})_{\mathcal{H}} + (\varepsilon(v_{0}), \varepsilon(\tilde{v}))_{L^{2}(\Omega; \Sigma)} + sb(\Phi_{0}, \tilde{\Phi}) = (\Psi, \tilde{\Phi})_{\mathcal{H}}, $$

where, thanks to [10] (Chapter 1, Theorem 2.7 and Corollary 2.9),

$$ \text{Ker } G_{0} = \left\{ \begin{pmatrix} E \\ v \end{pmatrix} \in \mathcal{H} \mid \text{curl } E = 0 \text{ in } \Omega \right\} = \nabla H_{\gamma}^{1}(\Omega) \times L^{2}(\Omega)^{3}. $$

with

$$ H_{\gamma}^{1}(\Omega) = \{\varphi \in H^{1}(\Omega)/\mathbb{R} \mid \varphi \text{ is constant along } \partial\Omega\}. $$

Therefore introducing $\varphi_{0} \in H_{\gamma}^{1}(\Omega)$ such that $E_{0} = \nabla\varphi_{0}$ and $u_{0} = v_{0}/s$, the variational formulation of the limit problem can be simplified. It reads: for $(J, F)^{T} \in \mathcal{H}$ given, find $(\varphi_{0}, u_{0}) \in H_{\gamma}^{1}(\Omega) \times H_{0}^{1}(\Omega)^{3}$ such that, for all, $(\tilde{\varphi}, \tilde{u}) \in H_{\gamma}^{1}(\Omega) \times H_{0}^{1}(\Omega)^{3}$,

$$ \left\{ \begin{array}{l} (\nabla\varphi_{0}, \nabla\tilde{\varphi})_{L^{2}(\Omega)^{3}} + (\varepsilon(u_{0}), \mathbf{d}\nabla\tilde{\varphi})_{L^{2}(\Omega)^{3}} = (s^{-2}J, \tilde{E})_{L^{2}(\Omega)^{3}}, \\ s^{2}(u_{0}, \tilde{v})_{L^{2}(\Omega)^{3}} + (\varepsilon(u_{0}), \varepsilon(\tilde{v}))_{L^{2}(\Omega; \Sigma)} - (\varepsilon(\tilde{v}), \mathbf{d}\nabla\varphi_{0})_{L^{2}(\Omega)^{3}} = (s^{-1}F, \tilde{v})_{L^{2}(\Omega)^{3}}. \end{array} \right. $$

These last equations are classically known as the quasi-static piezoelectric equations.
**5. Convergence estimates.** This section gives two convergence estimates when the strong or strict scale separation holds as well as an additional assumption on the image of $G_0$. These convergence estimates holds in the following equivalent norm in $V_0 = D(G_1)$

$$ \forall u \in V_0, \quad |||u|||^2 = \|su\|_{\mathcal{H}}^2 + \|G_1 u\|_{\mathcal{G}}^2 + \frac{|s|^2}{\eta} \Re e(b(u, u)). $$

We tackle first the easier case where the strict scale separation holds and conclude with the study of the strong scale separation case.

**5.1. Strict scale separation.**

**THEOREM 5.1.** Let the strict scale separation property holds and assume that the range of $G_0$ is closed, namely there exists $C_p > 0$ such that

$$ (5.1) \qquad \forall u \in \text{Ker } G_0^{\perp} \cap D(G_0), \quad \|u\|_{\mathcal{H}} \le C_p \|G_0 u\|_{\mathcal{G}}. $$

Assume, moreover, that there exists an anti-linear form $\ell_0 \in \mathcal{L}(D(G_1), \mathbb{C})$ such that, for all $\delta \in (0, 1)$, there exists $c_{\ell,\delta} \ge 0$ and $C_{\ell,\delta} \ge 0$ such that

$$ \forall v \in \text{Ker } G_0 \cap D(G_1), \quad |\ell_{\delta}(v) - \ell_0(v)| \le c_{\ell,\delta} \|v\|_{\mathcal{H}} + C_{\ell,\delta} \|G_1 v\|_{\mathcal{G}}. $$

Then, the sequence $\{u_\delta\} \subset D(G_0) \cap D(G_1)$ solution to (2.26), satisfies

$$ (5.2) \qquad (|||u_\delta - u_0|||^2 + \delta^{-2} \|G_0 u_\delta\|_{\mathcal{G}}^2)^{\frac{1}{2}} \le C_\delta $$

where

$$ C_{\delta} = \frac{1}{\eta} \max(c_{\ell,\delta}, |s|C_{\ell,\delta}, \delta C_0) \quad \text{with} \quad C_0 = |s|C_p(c_{\ell} + C_b \|su_0\|_{D(G_1)}). $$

Note that, when the source term is independent of $\delta$, i.e., $\ell_\delta = \ell_0$ in (2.26) then $c_{\ell,\delta} = C_{\ell,\delta} = 0$ and the obtain estimate shows that $u_\delta$ converges towards $u_0$ in $O(\delta)$.

*Proof.*

**Step 1: Decomposition of the solution.** Since $\text{Ker } G_0$ is a closed subspace of $\mathcal{H}$, the solution $u_\delta \in D(G_0) \cap D(G_1)$ to (2.26) can be decomposed as

$$ (5.3) \qquad u_{\delta} = u_{\delta,0} + z_{\delta}, $$

where $u_{\delta,0} \in \text{Ker } G_0$ and $z_\delta \in (\text{Ker } G_0)^\perp$ and the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. From the strict scale separation property,

$$ \overline{\text{Im } G_0^*} \perp \overline{\text{Im } G_1^*} \quad \Rightarrow \quad (\text{Ker } G_0)^\perp = \overline{\text{Im } G_0^*} \subset \text{Ker } G_1, $$

where the closure are taken for the norm in $\mathcal{H}$. This implies that $z_\delta \in \text{Ker } G_1$, and therefore

$$ u_{\delta,0} \in \text{Ker } G_0 \cap D(G_1) \quad \text{and} \quad z_\delta \in (\text{Ker } G_0)^\perp \cap \text{Ker } G_1. $$

**Step 2: Energy identity for the error equation.** The error term

$$ e_{\delta} = u_{\delta,0} - u_0 \in \text{Ker } G_0 \cap D(G_1), $$
is introduced. From the decomposition (5.3) it can be deduced that, for all $v \in$
$\mathrm{Ker} G_0 \cap D(G_1),$

$$
(5.4) \quad s^2(e_\delta, v)_H + (G_1 e_\delta, G_1 v) + sb(e_\delta + z_\delta, v) = \ell_\delta(v) - \ell_0(v).
$$

Choosing $v = se_\delta$ in (5.4) yields,

$$
(5.5) \quad s \|s e_{\delta}\|_{\mathcal{H}}^{2} + \bar{s} \|G_{1} e_{\delta}\|_{\mathcal{G}}^{2} + |s|^{2} b(e_{\delta} + z_{\delta}, e_{\delta}) = (\ell_{\delta}(se_{\delta}) - \ell_{0}(se_{\delta})).
$$

Choosing $v = sz_\delta$ in (2.26) yields, using the orthogonality of $e_\delta$ and $z_\delta$ in $\mathcal{H}$,

$$
(5.6) \quad s \|s z_{\delta}\|_{\mathcal{H}}^{2} + \bar{s} \delta^{-2} \|G_{0} z_{\delta}\|_{\mathcal{G}}^{2} + |s|^{2} b(u_{\delta,0} + z_{\delta}, z_{\delta}) = \ell_{\delta}(sz_{\delta}).
$$

**Step 3: Error estimate.** The next step is to sum (5.5) and (5.6), to get after taking
the real part,

$$
(5.7) \quad \|\|e_{\delta} + z_{\delta}\|\|^2 + \delta^{-2} \|G_0 z_{\delta}\|_{\mathcal{G}}^2 = \frac{1}{\eta} \Re e (\ell_{\delta}(se_{\delta}) - \ell_0(se_{\delta}) + \ell_{\delta}(sz_{\delta}) - b(su_0, sz_{\delta})).
$$

where, due to the properties of $e_\delta$ and $z_\delta$,

$$
\|\|e_{\delta} + z_{\delta}\|\|^2 = \|s e_{\delta}\|_{\mathcal{H}}^2 + \|s z_{\delta}\|_{\mathcal{H}}^2 + \|G_1 e_{\delta}\|_{\mathcal{G}}^2 + \frac{|s|^2}{\eta} \Re e (b(e_{\delta} + z_{\delta}, e_{\delta} + z_{\delta})).
$$

The three terms in the right hand side of Equation (5.7) are now estimated.

$$
(5.8) \quad \left| \left( \ell_{\delta}(se_{\delta}) - \ell_{0}(se_{\delta}) \right) \right| \le c_{\ell,\delta} \|se_{\delta}\|_{\mathcal{H}} + C_{\ell,\delta} |s| \|G_{1}e_{\delta}\|_{\mathcal{G}},
$$

moreover, since $z_{\delta} \in (\mathrm{Ker} G_0)^{\perp} \cap \mathrm{Ker} G_1$, and since the ranged of $G_0$ is closed,

$$
(5.9) \quad | \ell_{\delta}(sz_{\delta}) | \le c_{\ell} \|sz_{\delta}\|_{\mathcal{H}} \le c_{\ell} C_{p} |s| \|G_{0}z_{\delta}\|_{\mathcal{G}}.
$$

Finally, using the continuity property of $b$, the fact that $z_{\delta} \in \mathrm{Ker} G_1$ and again the
fact that the ranged of $G_0$ is closed,

$$
\begin{align*}
(5.10) \quad |s|^2 |b(u_0, z_\delta)| &\le C_b \|s u_0\|_{D(G_1)} \|s z_\delta\|_{D(G_1)} \\
&\le C_b C_p |s| \|s u_0\|_{D(G_1)} \|G_0 z_\delta\|_{\mathcal{G}}.
\end{align*}
$$

Summing (5.8), (5.9), (5.10), leads to,

$$
\begin{align*}
(5.11) \quad \mathcal{E}_{\delta} &:= \|\|e_{\delta} + z_{\delta}\|\|^2 + \delta^{-2} \|G_0 z_{\delta}\|_{\mathcal{G}}^2 \\
&\le \frac{c_{\ell,\delta}}{\eta} \|se_{\delta}\|_{\mathcal{H}} + |s| \frac{C_{\ell,\delta}}{\eta} \|G_1 e_{\delta}\|_{\mathcal{G}} + \frac{|s|}{\eta} C_p (c_{\ell} + C_b \|s u_0\|_{D(G_1)}) \|G_0 z_{\delta}\|_{\mathcal{G}}.
\end{align*}
$$

Hence, $\mathcal{E}_{\delta} \le C_{\delta} \mathcal{E}_{\delta}^{\frac{1}{2}}$ which concludes the proof. $\square$

<figure><img src="image_3.png" /></figure>

*Remark 5.2.* To apply Theorem 5.1 in the context of the piezo-electric equations presented Section 4.3 one has to verify (5.1). In this context we have

$$
\begin{align*}
\mathrm{Ker} G_0^{\perp} &= \left\{ \begin{pmatrix} E \\ v \end{pmatrix} \in L^2(\Omega)^3 \times L^2(\Omega)^3 \mid v = 0 \text{ and } (E, \nabla\varphi)_{L^2(\Omega)^3} = 0 \quad \forall \varphi \in H_\gamma^1(\Omega) \right\} \\
&= \mathcal{W}_0 \times \{0\},
\end{align*}
$$

where $\mathcal{W}_0 = \{E \in H_0(\text{curl}; \Omega) \mid \text{div}E = 0 \text{ in } \Omega\}$. Therefore, (5.1) correspond to verify that, there exits $C_p > 0$ such that,

$$
\forall E \in \mathcal{W}_0, \quad \|E\|_{L^2(\Omega)^3} \le C_p \|\text{curl } E\|_{L^2(\Omega)^3}.
$$

This result is given in [10] Section 3.4.
## 5.2. Strong scale separation.

For conciseness, the following notation is introduced

$$ (5.12) \quad \mathcal{V}_{\infty} = D(G_0) \cap D(G_1), \quad \forall v \in \mathcal{V}_{\infty} \quad \|v\|_{\mathcal{V}_{\infty}}^2 = \|v\|_{\mathcal{H}}^2 + \|G_0 v\|_{\mathcal{G}}^2 + \|G_1 v\|_{\mathcal{G}}^2. $$

The bounded operator $G_0 \in \mathcal{L}(\mathcal{V}_{\infty}, \mathcal{G})$ is defined by

$$ \forall v \in \mathcal{V}_{\infty}, \quad G_0 v = G_0 v, $$

The main theorem of this section relies on the assumption that the range of $G_0$ is closed. It can be shown to be equivalent to the so-called inf-sup condition (see [5] for more details).

**ASSUMPTION 5.3.** The range of $G_0$ is closed, namely there exists $C_c > 0$ such that

$$ (5.13) \quad \forall u \in (\text{Ker } G_0)^{\perp}, \quad \|u\|_{\mathcal{V}_{\infty}} \le C_c \|G_0 u\|_{\mathcal{G}}, $$

where the orthogonality holds for the scalar product in $\mathcal{V}_{\infty}$.

**THEOREM 5.4.** Let the strong scale separation property holds, as well as Assumption 5.3. Assume that there exists an anti-linear form $\ell_0 \in \mathcal{L}(D(G_1), \mathbb{C})$ such that, for all $\delta \in (0, 1)$, there exists $c_{\ell, \delta} \ge 0$ and $C_{\ell, \delta} \ge 0$ such that

$$ \forall v \in \text{Ker } G_0 \cap D(G_1), \quad |\ell_{\delta}(v) - \ell_0(v)| \le c_{\ell, \delta} \|v\|_{\mathcal{H}} + C_{\ell, \delta} \|G_1 v\|_{\mathcal{G}}. $$

Then, the sequence $\{u_{\delta}\} \subset D(G_0) \cap D(G_1)$ solution to (2.26), satisfies

$$ (5.14) \quad (|||u_{\delta} - u_0|||^2 + \delta^{-2} \|G_0 u_{\delta}\|_{\mathcal{G}}^2)^{\frac{1}{2}} \le C_{\delta} $$

where

$$ C_{\delta} = \frac{1}{\eta} \max(c_{\ell, \delta}, |s|C_{\ell, \delta}, \delta \|s g_0\|_{\mathcal{G}}), $$

and $g_0$ is the Lagrange multiplier associated to the mixed form of the limit problem: find $(u_0, g_0) \in \mathcal{V}_{\infty} \times \text{Im } G_0$ such that for all, $v \in \mathcal{V}_{\infty}$,

$$ (5.15) \quad \begin{cases} s^2(u_0, v)_{\mathcal{H}} + (G_1 u_0, G_1 v)_{\mathcal{G}} + sb(u_0, v) + (g_0, G_0 v)_{\mathcal{G}} = \ell_0(v), \\ G_0 u_0 = 0. \end{cases} $$

Remark that, – as for Theorem 5.1 – when the source term is independent of $\delta$, i.e., $\ell_{\delta} = \ell_0$ in (2.26) then $c_{\ell, \delta} = C_{\ell, \delta} = 0$ and the obtain estimate shows that $u_{\delta}$ converges towards $u_0$ in $O(\delta)$. Moreover, one can show that the mixed problem (5.15) is well-posed using – in particular – Assumption 5.3.

**Proof.**

**Step 1: Introduction of a Lagrange multiplier.** For all $v \in \mathcal{V}_{\infty}$ a functional $L \in \mathcal{L}(\mathcal{V}_{\infty}, \mathbb{C})$ is defined by

$$ (5.16) \quad L(v) = \ell_0(v) - s^2(u_0, v)_{\mathcal{H}} - (G_1 u_0, G_1 v)_{\mathcal{G}} - b(s u_0, v). $$
From the Riesz representation theorem there exists a unique element $\ell \in \mathcal{V}_\infty$ such that

$$ \forall v \in \mathcal{V}_\infty, \quad (\ell, v)_{\mathcal{V}_\infty} = L(v). $$

Observe now that, since (2.27) holds, $(\ell, v)_{\mathcal{V}_\infty} = 0$ for all

$$ v \in \mathrm{Ker} \, G_0 \cap D(G_1) = \mathrm{Ker} \, G_0 \cap D(G_1) = \mathrm{Ker} \, G_0. $$

Hence, $\ell \in (\mathrm{Ker} \, G_0)^\perp$ (here the orthogonality relation holds for the scalar product in $\mathcal{V}_\infty$). Now, observe that

$$ (\mathrm{Ker} \, G_0)^\perp = \overline{\mathrm{Im} \, G_0^*}. $$

Since the range of $G_0$ is closed by Assumption 5.3, the range of $G_0^*$ is closed, hence $\ell$ belongs to the range of $G_0^*$. More precisely,

$$ \ell \in \mathrm{Im} \, G_0^* \quad \Rightarrow \quad \exists g \in D(G_0^*) \text{ such that } \ell = G_0^* g. $$

The result above can be slightly improved. Since $G_0^*$ has closed range, there exists $C_0 > 0$ such that

$$ \forall g \in (\mathrm{Ker} \, G_0^*)^\perp, \quad \|g\|_G \le C_0 \|G_0^* g\|_{\mathcal{V}_\infty}, $$

showing that $G_0^*$ is injective on $(\mathrm{Ker} \, G_0^*)^\perp = \overline{\mathrm{Im} \, G_0} = \mathrm{Im} \, G_0$. Hence the following unique characterization of $\ell$ is deduced,

$$ \ell \in \mathrm{Im} \, G_0^* \quad \Rightarrow \quad \exists! g_0 \in \mathrm{Im} \, G_0 \text{ such that } \ell = G_0^* g_0. $$

Using (5.16) and the property of the adjoint

$$ \forall v \in \mathcal{V}_\infty, \quad L(v) = (\ell, v)_{\mathcal{V}_\infty} = (G_0^* g_0, v)_{\mathcal{V}_\infty} = (g_0, G_0 v)_G = (g_0, G_0 v)_G, $$

it has been shown that there exists a unique $g_0 \in \mathrm{Im} \, G_0$ such that (5.15) is satisfied.

**Step 2: Mixed formulation for the family of problems.** The objective is now to write a formulation for the family of problems that is similar to the one obtained just above. An element $g_\delta \in \mathrm{Im} \, G_0$ is introduced,

$$ g_\delta = \delta^{-2} G_0 u_\delta \in \mathrm{Im} \, G_0. $$

The solution to (2.26) is satisfies the variational formulation: for all $v \in \mathcal{V}_\delta$,

$$ (5.17) \qquad \begin{cases} s^2(u_\delta, v)_H + (G_1 u_\delta, G_1 v)_G + sb(u_\delta, v) + (g_\delta, G_0 v)_G = \ell_\delta(v), \\ G_0 u_\delta = \delta^2 g_\delta. \end{cases} $$

**Step 3: Energy identity for the error equation.** The following error terms are introduced,

$$ e_\delta = u_\delta - u_0 \in \mathcal{V}_\infty, \quad h_\delta = g_\delta - g_0 \in \mathrm{Im} \, G_0. $$

Using (5.15) and (5.17) one finds that: for all $v \in \mathcal{V}_\infty$,

$$ (5.18) \qquad \begin{cases} s^2(e_\delta, v)_H + (G_1 e_\delta, G_1 v) + sb(e_\delta, v)_G + (h_\delta, G_0 v)_G \\ = \ell_\delta(v) - \ell_0(v), \\ G_0 e_\delta = \delta^2 h_\delta + \delta^2 g_0. \end{cases} $$
Choosing $v = s e_\delta$, it comes

$$ (5.19) \quad s \|se_\delta\|_{\mathcal{H}}^2 + \bar{s}(G_1 e_\delta, G_1 e_\delta)_{\mathcal{G}} + |s|^2 b(e_\delta, e_\delta) + \bar{s}(h_\delta, G_0 e_\delta)_{\mathcal{G}} = (\ell_\delta(se_\delta) - \ell_0(se_\delta)). $$

The second equation of (5.18) can be used to simplify the identity just above, indeed,

$$ (5.20) \qquad (h_\delta, G_0 e_\delta)_{\mathcal{G}} = \delta^2 \|h_\delta\|_{\mathcal{G}}^2 + \delta^2 (h_\delta, g_0)_{\mathcal{G}}. $$

Using (5.20) in (5.19) the following energy identity can be derived,

$$ (5.21) \qquad |||e_\delta|||^2 + \delta^2 \|h_\delta\|_{\mathcal{G}}^2 = \frac{1}{\eta} \Re(e(\ell_\delta(se_\delta) - \ell_0(se_\delta) - \delta^2(h_\delta, sg_0))_{\mathcal{G}}), $$

leading to the estimation

$$ |||e_\delta|||^2 + \delta^2 \|h_\delta\|_{\mathcal{G}}^2 \le \frac{1}{\eta} (c_{\ell,\delta} \|se_\delta\|_{\mathcal{H}} + |s| C_{\ell,\delta} \|G_1 e_\delta\|_{\mathcal{G}}) + \frac{\delta}{\eta} \|sg_0\|_{\mathcal{G}} \delta \|h_\delta\|_{\mathcal{G}}. $$

Finally, one can deduce that $|||e_\delta|||^2 + \delta^2 \|h_\delta\|_{\mathcal{G}}^2 \le C_\delta^2$ with $C_\delta$ as defined in the statement of the Theorem. □

*Remark 5.5.* To apply Theorem 5.4 in the context of the nearly-incompressible elastodynamics equations presented Section 4.1 one has to verify Assumption 5.3. In this context, this corresponds to verify that the divergence operator from $V_\infty = H_0^1(\Omega)^3$ to $L^2(\Omega)$ has a closed ranged. This result is shown in [10] Section 2.2.
**6. Proofs of some of the results from Section 2 and Section 3.** The proofs below uses several results in operator theory that the reader can find in [11] Chap. 12, or [15] Chap 3. Section 5. These results are given below for convenience. Let

$$T: D(T) \subset \mathcal{H} \to \mathcal{G}, \quad S \subset D(S): \mathcal{H} \to \mathcal{G}, \quad R: D(R) \subset \mathcal{G} \to \mathcal{H}, \quad B \in \mathcal{L}(\mathcal{G}),$$

then

$$D(TR) = \{g \in D(R) \mid R g \in D(T)\}, \quad (T+S)R = TR + SR.$$

When T and S are densely defined then

$$(BT)^* = T^*B^* \quad \text{and} \quad T^* + S^* \subset (T+S)^*,$$

finally, when TR is densely defined, $R^*T^* \subset (TR)^*$.

**Proof of Theorem 2.3.** Only (2.4)(a) is proven, the property (2.4)(b) is proven similarly. Since $u_\delta$ is uniformly bounded in $\mathcal{H}$ it converges, up to a subsequence, weakly to $u_0$ in $\mathcal{H}$, for all $h \in D(G_0^*) \cap D(G_1^*)$ their holds,

$$(u_\delta, G_0^* h)_\mathcal{H} \xrightarrow{\delta \to 0} (u_0, G_0^* h)_\mathcal{H}.$$

When $(u_\delta, G_0^* h)_G \xrightarrow{\delta \to 0} 0$ it holds

$$(u_0, G_0^* h)_\mathcal{H} = 0, \quad \forall h \in D(G_0^*) \cap D(G_1^*).$$

By density of $D(G_0^*) \cap D(G_1^*)$ into $D(G_0^*)$ for the graph norm of $G_0^*$ it can be deduced that the equality above holds for all $h \in D(G_0^*)$. Since $G_0$ is closed and densely defined, this shows that $u_0 \in D(G_0)$ and $G_0 u_0 = 0$.

**Proof of Lemma 2.5.** Thanks to (2.3), the operator $\delta^{-1}G_0 + G_1$ is densely defined and closable, its adjoint satisfies

$$(\delta^{-1}G_0^* + G_1^*) \subset (\delta^{-1}G_0 + G_1)^* = \overline{\delta^{-1}G_0 + G_1}^* = G_\delta^*$$

Therefore,

$$(\delta^{-1}G_0^* + G_1^*) Q_0 = \delta^{-1}G_0^* Q_0 + G_1^* Q_0 = G_1^* Q_0 \subset G_\delta^* Q_0 = (Q_0 G_\delta)^*,$$

hence, summarizing,

$$(6.1) \qquad G_1^* Q_0 \subset (Q_0 G_\delta)^*.$$

However, since $Q_0G_1$ is closable, $G_1^* Q_0$ is densely defined (Lemma 3.1), it can then be deduced from (6.1) that $(Q_0 G_\delta)^*$ is also densely defined and therefore $Q_0 G_\delta$ is closable. Moreover from (6.1), and the property that $Q_0$ is bounded and self-adjoint, on can write

$$\overline{Q_0 G_\delta} = (Q_0 G_\delta)^{**} \subset (G_1^* Q_0)^* = (Q_0 G_1)^{**} = \overline{Q_0 G_1},$$

which conclude the proof.
**Proof of Theorem 2.12.** The objective is to prove that

$$ egin{cases} Q_0 G_1 = G_1 \\ D(G_0^*) \cap D(G_1^*) \text{ dense in } D(G_0^*), \\ \delta^{-1}G_0 + G_1 \text{ is closed.} \end{cases} $$

The other statements of the theorem are direct consequences of the above properties.

**The equality** $Q_0 G_1 = G_1$. By assumption,

$$ \forall (u, v) \in D(G_0) \times D(G_1) \quad (G_0 u, G_1 v)_G = 0. $$

By density, this, implies

$$ \forall (g, v) \in \overline{\mathrm{Im}(G_0)} \times D(G_1) \quad (g, G_1 v)_G = 0, $$

where $\overline{\mathrm{Im}(G_0)}$ is the closure of the range of $G_0$ for the norm in $\mathcal{G}$. Now, since $I - Q_0$ is, by definition, the orthogonal projection operator on the closure of $\mathrm{Im} G_0$, there holds

$$ \forall (g, v) \in \mathcal{G} \times D(G_1) \quad ((I - Q_0)g, G_1 v)_G = 0, $$

also meaning that, for all $v \in D(G_1)$, $G_1 v = Q_0 G_1 v$.

**Density results.** It has been shown that $Q_0 G_1 = G_1$ hence $Q_0 G_1$ is closed and densely defined. Therefore $(Q_0 G_1)^* = G_1^* G_0^*$ is closed and densely defined in $\mathcal{G}$. More precisely,

$$ D(G_1^* Q_0) = (\mathrm{Ker} G_0^* \cap D(G_1^*)) \oplus (\mathrm{Ker} G_0^*)^\perp \text{ dense in } \mathcal{G} = \mathrm{Ker} G_0^* \oplus (\mathrm{Ker} G_0^*)^\perp. $$

From the statement it can be deduced that $\mathrm{Ker} G_0^* \cap D(G_1^*)$ is dense in $\mathrm{Ker} G_0^*$. It must be check now that $D(G_0^*) \cap D(G_1^*)$ is dense in $D(G_0^*)$ for the graph norm of $G_0^*$. The following vectorial space is defined,

$$ \mathcal{Z} = (\mathrm{Ker} G_0^* \cap D(G_1^*)) \oplus ((\mathrm{Ker} G_0^*)^\perp \cap D(G_0^*)), $$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{G}$. Observe that $\mathcal{Z}$ is a subspace of $D(G_0^*) \cap D(G_1^*)$ since by assumption

$$ (\mathrm{Ker} G_0^*)^\perp = \overline{\mathrm{Im} G_0} \subset \mathrm{Ker} G_1^*. $$

Observe, moreover, that

$$ D(G_0^*) = \mathrm{Ker} G_0^* \oplus ((\mathrm{Ker} G_0^*)^\perp \cap D(G_0^*)), $$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{G}$. However, since it has been shown that $\mathrm{Ker} G_0^* \cap D(G_1^*)$ is dense in $\mathrm{Ker} G_0^*$ for the norm $\mathcal{H}$, it is therefore dense for the graph norm of $G_0^*$. This shows that $\mathcal{Z}$ (hence $D(G_0^*) \cap D(G_1^*)$) is dense in $D(G_0^*)$ for the same norm.

**The operator $\delta^{-1}G_0 + G_1$ is closed.** The graph of the operator $\delta^{-1}G_0 + G_1$ is given by

$$ \mathcal{G}_r(\delta^{-1}G_0 + G_1) = \{(u, g) \in \mathcal{H} \times \mathcal{G} \mid u \in D(G_0) \cap D(G_1), g = \delta^{-1}G_0 u + G_1 u\}. $$
It must be verified that the graph is closed for the composite norm in $\mathcal{H} \times \mathcal{G}$. Let $\{u_n\}$ be a sequence in $D(G_0) \cap D(G_1)$ such that, for some $(u,g) \in \mathcal{H} \times \mathcal{G}$,

$$u_n \to_{\mathcal{H}} u, \quad \delta^{-1}G_0u_n + G_1u_n \to_{\mathcal{G}} g.$$

This implies

$$Q_0G_1u_n = G_1u_n \to_{\mathcal{G}} Q_0g.$$

However, since $G_1$ is closed, $u \in D(G_1)$ and $Q_0g = G_1u$. Consequently

$$u_n \to_{\mathcal{H}} u, \quad \delta^{-1}G_0u_n \to_{\mathcal{G}} (I - Q_0)g,$$

and since $G_0$ is closed, $u \in D(G_0)$ and $(I - Q_0)g = \delta^{-1}G_0u$ which conclude the proof since

$$u \in D(G_0) \cap D(G_1) \text{ and } g = \delta^{-1}G_0u + G_1u.$$

**Proof of Theorem 2.14.** Our objective is to prove that

$$D(G_0) \cap D(G_1) \text{ dense in } D(G_0),$$

using that, by assumption, $\overline{\mathrm{Im}(G_1^*)} \perp \overline{\mathrm{Im}(G_0^*)}$, which implies that

$$ (6.2) \qquad \begin{cases} (\mathrm{Ker} G_1)^\perp = \overline{\mathrm{Im}(G_1^*)} \subset \mathrm{Ker} G_0, & (a) \\ (\mathrm{Ker} G_0)^\perp = \overline{\mathrm{Im}(G_0^*)} \subset \mathrm{Ker} G_1. & (b) \end{cases} $$

As a direct consequence of (6.2)(a) it can be deduced that

$$P_0G_1^* = G_1^* \Rightarrow G_1P_0 = G_1,$$

where $P_0$ is the orthogonal projection operator on $\mathrm{Ker} G_0$. Therefore $D(G_1P_0) = \mathrm{Ker} G_0 \cap D(G_1) \oplus (\mathrm{Ker} G_0)^\perp$ is dense in $\mathcal{G}$, which implies that

$$ (6.3) \qquad \mathrm{Ker} G_0 \cap D(G_1) \text{ dense in } \mathrm{Ker} G_0. $$

One defines the space

$$\mathcal{W} = (\mathrm{Ker} G_0 \cap D(G_1)) \oplus ((\mathrm{Ker} G_0)^\perp \cap D(G_0)),$$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. Observe that the space $\mathcal{W}$ is a subspace of $D(G_0) \cap D(G_1)$ thanks to (6.2)(b). Moreover,

$$D(G_0) = \mathrm{Ker} G_0 \oplus ((\mathrm{Ker} G_0)^\perp \cap D(G_0)),$$

where the decomposition is orthogonal with respect to the scalar product in $\mathcal{H}$. It can be concluded that $\mathcal{W}$ is dense in $D(G_0)$ for the graph norm of $G_0$ using (6.3).
**Proof of Lemma 3.1.** First note that $Q_0G_1$ is densely defined hence if $Q_0G_1$ is closable then $(Q_0G_1)^*$ is densely defined, i.e., since $Q_0$ is bounded and self-adjoint,

$$(Q_0G_1)^* = G_1^*Q_0$$

is densely defined. Reciprocally, if $G_1^*Q_0$ is densely defined it has an adjoint

$$(G_1^*Q_0)^* \supset Q_0G_1^{**} = Q_0G_1,$$

hence $Q_0G_1$ is closable since it has a closed extension. Left to show that $G_1^*Q_0$ is densely defined if and only if

$$(6.4) \qquad \text{Ker } G_0^* \cap D(G_1^*) \text{ dense in } (\text{Ker } G_0^*; \| \cdot \|_G).$$

One has,

$$D(G_1^* Q_0) = \{g \in D(Q_0) = \mathcal{G} \mid Q_0 g \in D(G_1^*)\}.$$

For all $g \in \mathcal{G}$, an orthogonal decomposition in $\mathcal{G}$ holds. It is given by $g = g_0 + g_\perp$ with $g_0 \in \text{Ker } G_0^*$ and $Q_0 g_\perp = 0$. Therefore

$$D(G_1^* Q_0) = \{g = g_0 + g_\perp, g_0 \in \text{Ker } G_0^*, g_\perp \in (\text{Ker } G_0^*)^\perp \mid Q_0 g = g_0 \in D(G_1^*)\},$$

which shows that $D(G_1^* Q_0) \subset (\text{Ker } G_0^* \cap D(G_1^*)) \oplus (\text{Ker } G_0^*)^\perp$. The reverse inclusion being straightforward to prove we have

$$(6.5) \qquad D(G_1^* Q_0) = (\text{Ker } G_0^* \cap D(G_1^*)) \oplus (\text{Ker } G_0^*)^\perp,$$

which indeed shows that (6.4) is equivalent to $G_1^* Q_0$ densely defined.

## REFERENCES

[1] G. Allaire. Shape optimization by the homogenization method. Springer, 2002.

[2] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer-Verlag New York, 1983.

[3] A. Bensoussan and J.L. Lions. Asymptotic analysis for periodic structures. North Holland, 1978.

[4] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter. Representation and Control of Infinite Dimensional Systems (Systems & Control: Foundations & Applications). Birkhauser, 2006.

[5] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer, 1991.

[6] F. Caforio and S. Imperiale. Mathematical analysis of a penalization strategy for incompressible elastodynamics. Asymptotic Analysis, 122(3-4), 2020.

[7] D. Chapelle and K.-J. Bathe. The finite element analysis of shells - fundamentals. Computational Fluid and Solid Mechanics. Springer-Verlag Berlin Heidelberg, 2011.

[8] P. G. Ciarlet. Plates and junctions in elastic multi-structures : an asymptotic analysis. Recherches en matheématiques appliquées 14. Masson, Springer-Verlag, 1990.

[9] R. Dautray and J.L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I. Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag Berlin Heidelberg, 1992.

[10] V. Girault and P.-A. Raviart. Finite element approximation of the Navier-Stokes equations. Technical report, Springer-Verlag, 1979.

[11] G. Grubb. Distributions and Operators. Graduate Texts in Mathematics. Springer New York, 2008.

[12] D. Huet. Phénomènes de perturbation singulière dans les problèmes aux limites. Annales de l'Institut Fourier, 10, 1960.

[13] S. Imperiale and P. Joly. Mathematical and numerical modelling of piezoelectric sensors. ESAIM: Mathematical Modelling and Numerical Analysis, 46, 2012.
[14] S. Imperiale and P. Joly. *Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section*. Applied Numerical Mathematics, 79, 2013.

[15] T. Kato. *Perturbation Theory for Linear Operators*. Classics in Mathematics. Springer Berlin Heidelberg, 1995.

[16] J.L. Lions. *Quelques méthodes de résolution des problèmes aux limites non linéaires*. Paris, Dunod, 1969.

[17] J.L. Lions and E. Magenes. *Problèmes aux limites non homogènes et Applications Vol. 1*. Dunod, 1968.

[18] R. Temam. *Navier-Stokes Equations: Theory and Numerical Analysis*, Studies in mathematics and its application, North-Holland Publishing Company, 1979.

[19] M. Tucsnak and G. Weiss. *Observation and Control for Operator Semigroups*. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser Basel, 2009.

[20] G. Allaire. *Homogenization and two-scale convergence*. SIAM Journal on Mathematical Analysis, 23(6), 1992.

[21] D. Cioranescu, A. Damlamian and G. Griso. *The Periodic Unfolding Method*. Series in Contemporary Mathematics, Springer Nature, 2018.

[22] A. Braides and A. Defranceschi. *Homogenization of Multiple Integrals*. Oxford University Press, Oxford, 1998

[23] M. Waurick. *Homogenization of a class of linear partial differential equations* Asymptotic Analysis, 82(3-4), 2013.

[24] B. Cockburn and P. Joly. *Maxwell equations in polarizable media*. SIAM Journal on Mathematical Analysis, 19(6), 1988.

[25] G. Cohen, P. Joly, J. E. Roberts and N. Tordjman. *High order triangular finite elements with mass lumping for the wave equation*. SIAM Journal on Numerical Analysis, 38(6), 2001.

[26] P. Amenoagbadji, S. Fliss and P. Joly. *Wave propagation in one-dimensional quasiperiodic media*. arXiv, 2023.

[27] S. Cooper and I. Kamotski and V. P. Smyshlyaev, *Uniform asymptotics for a family of degen- erating variational problems and multiscale approximations with error estimates*. arXiv, 2023

[28] M. E. Gurtin. *The Linear Theory of Elasticity*. Springer Berlin Heidelberg, 1973

[29] M. Sh. Birman and T. A. Suslina. *Second order periodic differential operators. Threshold properties and homogenisation*. St. Petersburg Mathematical journal, 15, 2004.

[30] V. V. Zhikov and S. E. Pastukhova. *Operator estimates in homogenization theory*. Russian Mathematical Surveys, 71(3), 2016.

[31] S. E. Pastukhova. *Homogenization estimates for singularly perturbed operators*. Journal of Mathematical Sciences, 251(5), 2020.

[32] A. Piatnitski and V. Slousch and T. Suslina and E. Zhizhina *On operator estimates in homogenization of nonlocal operators of convolution type*. Journal of Differential Equations, 352, 2023.