Numerical multiscale methods for Maxwell's equations in heterogeneous media

In this thesis we introduce new numerical multiscale methods for problems arising from time-harmonic Maxwell's equations in heterogeneous media. These problems are used to model electromagnetic wave propagation, for instance, in the context of photonic crystals. Such materials can exhibit unusual optical properties, where we are in particular interested in negative refraction. Although this phenomenon and its effects have been studied in a lot of physical experiments, the mathematical understanding of this topic is still in its infancy. As a first step, we consider elliptic problems with heterogeneous (rapidly varying) coefficients involving the double application of the curl as differential operator. The corresponding solutions typically admit very low regularity and conventional numerical schemes have arbitrarily bad convergence rates. For locally periodic problems, we suggest a Heterogeneous Multiscale Method and prove a priori error estimates.Numerical experiments are given to confirm the convergence rates and to validate the applicability of the method. In order to cope with more general coefficients, we construct a generalized finite element method in the spirit of the Localized Orthogonal Decomposition. The method decomposes the exact solution into a coarse-scale part (spanned by standard finite element functions) and a fine-scale part. A stable corrector operator, which is quasi-local and thus can be computed efficiently, allows to represent and extract necessary fine-scale features of the solution. We show that this construction enjoys optimal approximation properties in energy and dual norms. As the next and even more challenging step towards negative refraction, we consider (indefinite) scattering problems with periodic high contrast coefficients. Here, periodically distributed inclusions are associated with a much smaller material coefficient (scaled like the square of the periodicity length) than the rest of the scatterer. Homogenization results show that the high contrast leads to unusual effective parameters in the homogenized equation. Consequently, wave propagation inside the scatterer is physically forbidden for certain wavenumbers; this effect is called a band gap. In the analysis of the homogenized formulation, we particularly prove new wavenumber-explicit stability estimates for solutions to the Helmholtz and Maxwell equations. As numerical discretization scheme we propose a Heterogeneous Multiscale Method, for which we show inf-sup stability, quasi-optimality, and a priori error estimates. These results are obtained under a (standard) resolution condition between the wavenumber and the mesh size. Numerical experiments confirm the convergence rates and give an explanation of the physical phenomenon of band gaps. The whole PhD thesis is available here.