Homogenisation for Maxwell and Friends

TL;DR

This paper investigates the continuous dependence and convergence properties of solution operators in homogenisation problems for Maxwell's equations, highlighting operator-theoretic insights applicable to nonlocal and oscillatory coefficients.

Contribution

It refines the understanding of solution operator convergence under nonlocal topologies and provides sharp examples, including numerical and analytic treatments, for Maxwell's equations with oscillating coefficients.

Findings

Certain solution operator components converge strongly.

Weak convergence behavior aligns with known homogenisation results.

Numerical and analytic examples demonstrate sharpness of theoretical results.

Abstract

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.