There has been considerable recent interest in composite materials whose macroscopic physical properties can be radically different from those of conventional materials, often due to effects of the so-called "micro-resonances". Mathematically this leads to studying high-contrast homogenization of (periodic or not) problems with a "critically" scaled high contrast, where the resulting two-scale asymptotic behaviour appears to display a number of interesting effects. Mathematical analysis of these problems requires development of "two-scale" versions of operator and spectral convergences, of compactness, etc. We will review some background, as well as some more recent developmens and applications. One is two-scale analysis of general "partially-degenerating" periodic PDE problems [1], where strong two-scale resolvent convergence appears to hold under a rather generic decomposition assumption, implying in particular (two-scale) convergence of semigroups with applications to a wide class of micro-resonant dynamic problems.

A substantial additional effort is required for establishing not only the convergence but also its rate i.e. error bounds. In [2], we establish such error bounds for eigenvalue and eigenmodes due to a localized defect in a high-contrast periodic medium, which mathematical problem is motivated in particular by formation of localized modes in photonic crystal fibers.

Finally we will briefly review a most recent generic approach for establishing operator error bounds, i.e. error estimates in the strongest possible sense, for high-contrast infinite periodic problems [3]. The latter is achieved for a rather general class of high-contrast periodic problems via the use of Floquet-Bloch theory and careful analysis of related quadratic forms and of projection operators on relevant "macro" and "micro" subspaces and their interactions. This allows to establish the desired operator-type error bounds, and as one particular implication to establish error bounds on the spectral convergence including on the spectral band gaps, which can be illustrated by some explicit examples.
Various parts of the work are joint with Ilia Kamotski and Shane Cooper.

[1] I.V. Kamotski, V.P. Smyshlyaev, Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients, to appear in Applicable Analysis (2018), published online February 2018. Available also as https://arxiv.org/pdf/1309.4579v2.pdf
[2] I. V. Kamotski, V. P. Smyshlyaev Localised modes due to defects in high contrast periodic media via two-scale homogenization, to appear in J. Math. Sci. (NY) (2018). Available also as https://arxiv.org/pdf/1801.03372v1.pdf
[3] S. Cooper, I.V. Kamotski, V.P. Smyshlyaev. To be published, 2018.