  If an elliptic differential operator associated with an
$\mathbf{H}(\mathrm{curl})$-problem involves rough (rapidly varying)
coefficients, then solutions to the corresponding
$\mathbf{H}(\mathrm{curl})$-problem admit typically very low regularity, which
leads to arbitrarily bad convergence rates for conventional numerical schemes.
The goal of this paper is to show that the missing regularity can be
compensated through a corrector operator. More precisely, we consider the
lowest order N\'ed\'elec finite element space and show the existence of a
linear corrector operator with four central properties: it is computable,
$\mathbf{H}(\mathrm{curl})$-stable, quasi-local and allows for a correction of
coarse finite element functions so that first-order estimates (in terms of the
coarse mesh-size) in the $\mathbf{H}(\mathrm{curl})$ norm are obtained provided
the right-hand side belongs to $\mathbf{H}(\mathrm{div})$. With these four
properties, a practical application is to construct generalized finite element
spaces which can be straightforwardly used in a Galerkin method. In particular,
this characterizes a homogenized solution and a first order corrector,
including corresponding quantitative error estimates without the requirement of
scale separation.
