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In this article we provide a method for establishing operator-type error estimates between solutions to rapidly oscillating evolutionary equations and their homogenised counter parts. This method is exemplified by applications to the wave, heat and finally thermoelastic evolutionary systems.

We study an abstract family of asymptotically degenerating variational problems. Those are natural generalisations of families of problems emerging upon application of a rescaled Floquet-Bloch-Gelfand transform to resolvent problems for high-contrast elliptic PDEs with highly oscillatory periodic coefficients. An asymptotic analysis of these models leads us to a hierarchy of approximation results with uniform operator-type error estimates under various assumptions, satisfied by specific examples. We provide approximations for the resolvents in terms of a certain `bivariate' operator which appears an abstract generalisation of the two-scale limit operators for highly oscillatory high-contrast PDEs. The resulting approximating self-adjoint operator, providing tight operator error estimates, is the bivariate operator sandwiched by a connecting operator which for a broad class of periodic problems specialises to a new two-scale version of the classical Whittaker-Shannon interpolation. An explicit description of the limit spectrum in the abstract setting is provided, and new tight error estimates on the distance between the original and limit spectra are established. Our generic approach allows us to readily consider a wide class of asymptotically degenerating problems including but also going beyond high-contrast highly oscillatory PDEs. The obtained results are illustrated by various examples.

Homogenisation of global $\mathcal{A}^\epsilon$ and exponential $\mathcal{M}^\epsilon$ attractors for the damped semi-linear anisotropic wave equation $\partial_t^2 u^\epsilon +\gamma\partial_t u^\epsilon-{\rm div} \left(a\left( \tfrac{x}{\epsilon} \right)\nabla u^\epsilon \right)+f(u^\epsilon)=g$, on a bounded domain $\Omega \subset \mathbb{R}^3$, is performed. Order-sharp estimates between trajectories $u^\epsilon(t)$ and their homogenised trajectories $u^0(t)$ are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator ${\rm div}\left(a\left( \tfrac{x}{\epsilon} \right)\nabla \right)$ and its homogenised limit ${\rm div}\left(a^h\nabla \right)$. Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts $\mathcal{A}^0$ and $\mathcal{M}^0$ are established. These results imply error estimates of the form ${\rm dist}_X(\mathcal{A}^\epsilon, \mathcal{A}^0) \le C \epsilon^\varkappa$ and ${\rm dist}^s_X(\mathcal{M}^\epsilon, \mathcal{M}^0) \le C \epsilon^\varkappa$ in the spaces $X =L^2(\Omega)\times H^{-1}(\Omega)$ and $X =(C^\beta(\overline{\Omega}))^2$. In the natural energy space $\mathcal{E} : = H^1_0(\Omega) \times L^2(\Omega)$, error estimates ${\rm dist}_{\mathcal{E}}(\mathcal{A}^\epsilon, {T}_\epsilon \mathcal{A}^0) \le C \sqrt{\epsilon}^\varkappa$ and ${\rm dist}^s_{\mathcal{E}}(\mathcal{M}^\epsilon, {T}_\epsilon \mathcal{M}^0) \le C \sqrt{\epsilon}^\varkappa$ are established where ${T}_\epsilon$ is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by `fibre-wise' homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods., Comment: 38 pages

Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\l_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely" localised to the defect, in the sense that the localisation exponent behaves as $\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing literature. As a consequence, we argue that $\l_0$ is an eigenvalue of a certain limit operator defined on the defect only. In two- and three-dimensional configurations, whose one-dimensional cross-sections are described by the setting considered, this implies the existence of propagating waves that are localised to the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasimomentum dependence are found as resolvent limits of the original operator family, resulting in limiting spectral behaviour with a rich dependence on quasimomenta.

We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a "macroscopic" domain $(0,T)^d, T>0.$ We consider the case when the contrast between the values of the coefficients in different parts of their periodicity cell increases as the period of oscillations $\eta$ goes to zero. We show that the limit of the spectrum as $\eta\to0$ contains the spectrum of a "homogenised" system of equations that is solved by the limits of sequences of eigenfunctions of the original problem. We investigate the behaviour of this system and demonstrate phenomena not present in the scalar theory for polarised waves.

We consider a heterogeneous elastic structure which is stratified in some direction. We derive the limit problem under the assumption that the Lam\'e coefficients and their inverses weakly* converge to Radon measures. Our method applies also to linear second-order elliptic systems of partial differential equations and in particular, for the case $d=1$, this addresses the previously open problem of determining the asymptotic behaviour in this context for the general anisotropic heat equation.

We consider the Maxwell's system for a periodic array of dielectric `fibers' embedded into a `matrix', with respective electric permittivities $\epsilon_0$ and $\epsilon_1$, which serves as a model for cladding in photonic crystal fibers (PCF). The interest is in describing admissible and forbidden (gap) pairs $(\omega,k)$ of frequencies $\omega$ and propagation constants $k$ along the fibers, for a Bloch wave solution on the cross-section. We show that, for "pre-critical" values of $k(\omega)$ i.e. those just below $\omega (\min\{\epsilon_0,\epsilon_1\}\mu)^{1/2}$ (where $\mu$ is the magnetic permeability assumed constant for simplicity), the coupling specific to the Maxwell's systems leads to a particular partially degenerating PDE system for the axial components of the electromagnetic field. Its asymptotic analysis allows to derive the limit spectral problem where the fields are constrained in one of the phases by Cauchy-Riemann type relations. We prove related spectral convergence. We finally give some examples, in particular of small size "arrow" fibers ($\epsilon_0>\epsilon_1$) where the existence of the gaps near appropriate "micro-resonances" is demonstrated by a further asymptotic analysis., Comment: 36 pages, 1 figure

We study the asymptotic behaviour of the resolvents $({\mathcal A}^\varepsilon+I)^{-1}$ of elliptic second-order differential operators ${\mathcal A}^\varepsilon$ in ${\mathbb R}^d$ with periodic rapidly oscillating coefficients, as the period $\varepsilon$ goes to zero. The class of operators covered by our analysis includes both the "classical" case of uniformly elliptic families (where the ellipticity constant does not depend on $\varepsilon$) and the "double-porosity" case of coefficients that take contrasting values of order one and of order $\varepsilon^2$ in different parts of the period cell. We provide a construction for the leading order term of the "operator asymptotics" of $({\mathcal A}^\varepsilon+I)^{-1}$ in the sense of operator-norm convergence and prove order $O(\varepsilon)$ remainder estimates., Comment: 20 pages

In this chapter we describe a selection of mathematical techniques and results that suggest interesting links between the theory of gratings and the theory of homogenization, including a brief introduction to the latter. By no means do we purport to imply that homogenization theory is an exclusive method for studying gratings, neither do we hope to be exhaustive in our choice of topics within the subject of homogenization. Our preferences here are motivated most of all by our own latest research, and by our outlook to the future interactions between these two subjects. We have also attempted, in what follows, to contrast the "classical" homogenization (Section 11.1.2), which is well suited for the description of composites as we have known them since their advent until about a decade ago, and the "non-standard" approaches, high-frequency homogenization (Section 11.2) and high-contrast homogenization (Section 11.3), which have been developing in close relation to the study of photonic crystals and metamaterials, which exhibit properties unseen in conventional composite media, such as negative refraction allowing for super-lensing through a flat heterogeneous lens, and cloaking, which considerably reduces the scattering by finite size objects (invisibility) in certain frequency range. These novel electromagnetic paradigms have renewed the interest of physicists and applied mathematicians alike in the theory of gratings.

A two phase elastic composite with weakly compressible elastic inclusions is considered. The homogenised two-scale limit problem is found, via a version of the method of two-scale convergence, and analysed. The microscopic part of the two-scale limit is found to solve a Stokes type problem and shown to have no microscopic oscillations when the composite is subjected to body forces that are microscopically irrotational. The composites spectrum is analysed and shown to converge, in an appropriate sense, to the spectrum of the two-scale limit problem. A characterisation of the two-scale limit spectrum is given in terms of the limit macroscopic and microscopic behaviours.