  In the analysis of the $h$-version of the finite-element method (FEM), with
fixed polynomial degree $p$, applied to the Helmholtz equation with wavenumber
$k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is
sufficiently small and the sequence of Galerkin solutions are quasioptimal;
here $C_{\rm sol}$ is the norm of the Helmholtz solution operator, normalised
so that $C_{\rm sol} \sim k$ for nontrapping problems. In the
$\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ is
sufficiently small, then (for physical data) the relative error of the Galerkin
solution is controllably small. In this paper, we prove the natural error
bounds in the preasymptotic regime for the variable-coefficient Helmholtz
equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or
combinations of these) and with the radiation condition $\textit{either}$
realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball
$\textit{or}$ approximated either by a radial perfectly-matched layer (PML) or
an impedance boundary condition. Previously, such bounds for $p>1$ were only
available for Dirichlet obstacles with the radiation condition approximated by
an impedance boundary condition. Our result is obtained via a novel
generalisation of the "elliptic-projection" argument (the argument used to
obtain the result for $p=1$) which can be applied to a wide variety of abstract
Helmholtz-type problems.
