# Fibre Homogenisation

## TL;DR

This paper introduces a novel fibre-wise homogenisation method for analyzing the asymptotic behaviour of resolvents in parameter-dependent PDE systems with rapidly oscillating coefficients, providing sharper error estimates.

## Contribution

It presents a new fibre homogenisation approach that improves understanding of the asymptotic behaviour of operator resolvents in complex PDE systems.

## Key findings

- Fibre homogenisation yields order-sharp error estimates.
- Fibre-homogenised resolvents are asymptotically equivalent to standard homogenisation results.
- Method applies to Maxwell and second-order PDE systems.

## Abstract

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by `fibre-wise' homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.