# Nonlocal homogenisation theory for curl-div-systems

## TL;DR

This paper develops a nonlocal homogenisation theory for curl-div systems with variable coefficients, addressing highly oscillatory behaviors and boundary conditions, extending existing convergence concepts in mathematical physics.

## Contribution

It introduces a new topology for nonlocal, non-periodic coefficients in curl-div systems, refining nonlocal H-convergence techniques for complex boundary conditions.

## Key findings

- Established a nontrivial limit system for standard boundary conditions.
- Analyzed highly oscillatory local coefficients in Maxwell's equations.
- Linked abstract homogenisation results to local H-convergence and weak* convergence.

## Abstract

We study the curl-div-system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal $H$-convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non-periodic coefficients in curl-div systems to model highly oscillatory behaviour of the coefficients on small scales. We address curl-div systems under various boundary conditions and analyse the limit of the ratio of small scale over large scale tending to zero. Already for standard Dirichlet boundary conditions and local coefficients the limit system is nontrivial and unexpected. Furthermore, we provide an analysis of highly oscillatory local coefficients for a curl-div system with impedance type boundary conditions relevant in scattering theory for Maxwell's equations and relate the abstract findings to local $H$-convergence and weak$*$-convergence of the coefficients.