I am broadly interested in analysis and applications of “multi-scale” partial differential equations (PDEs). The application areas range from mechanics of composite materials to propagation and scattering of waves (acoustic, electromagnetic or elastic), linking to further industrial applications.

In “non-classical homogenisation” the strong coupling between scales leads to asymptotically explicit multi-scale descriptions of such effects as localisation and dispersion of waves due to the so-called micro-resonances. The latter is one of the signatures of “metamaterials”, the composites with highly unusual physical properties. Work on new hybrid asymptotic-numerical methods in high-frequency scattering combines asymptotic approaches (both of geometro-optical and of “diffractive” nature) with those numerical by incorporating the oscillatory “ansatz” parts into discretised elements and hence efficiently computing the oscillatory solutions.

This all prompts development of a new analysis requiring revisiting in this multi-scale context of such fundamental concepts as convergence and compactness (in the spectral, operator, and variational theories). This analysis is of interest in its own right, but also assists in assessing the accuracy of the asymptotic models and of related numerical schemes.