Homogenization for Maxwell and Friends
Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick · Multiscale Modeling and Simulation · 2025
Abstract. We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal [Formula: see text]-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 behavior known from homogenization 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 also provided. 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.