  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.
