  For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz
equation, the question of how quickly the meshwidth $h$ must decrease with the
frequency $k$ to maintain accuracy as $k$ increases has been studied since the
mid 80's. Nevertheless, there still do not exist in the literature any
$k$-explicit bounds on the relative error of the FEM solution (the measure of
the FEM error most often used in practical applications), apart from in one
dimension. The main result of this paper is the sharp result that, for the
lowest fixed-order conforming FEM (with polynomial degree, $p$, equal to one),
the condition "$h^2 k^3$ sufficiently small" is sufficient for the relative
error of the FEM solution in 2 or 3 dimensions to be controllably small
(independent of $k$) for scattering of a plane wave by a nontrapping obstacle
and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds
on the FEM solution for arbitrary fixed-order methods applied to scattering by
a nontrapping obstacle, but these bounds are not sharp for $p\geq 2$. A key
ingredient in our proofs is a result describing the oscillatory behaviour of
the solution of the plane-wave scattering problem, which we prove using
semiclassical defect measures.
