  The high-frequency Helmholtz equation on the entire space is truncated into a
bounded domain using the perfectly matched layer (PML) technique and
subsequently, discretized by the higher-order finite element method (FEM) and
the continuous interior penalty finite element method (CIP-FEM). By formulating
an elliptic problem involving a linear combination of a finite number of
eigenfunctions related to the PML differential operator, a wave-number-explicit
decomposition lemma is proved for the PML problem, which implies that the PML
solution can be decomposed into a non-oscillating elliptic part and an
oscillating but analytic part. The preasymptotic error estimates in the energy
norm for both the $p$-th order CIP-FEM and FEM are proved to be $C_1(kh)^p +
C_2k(kh)^{2p} +C_3 E^{\rm PML}$ under the mesh condition that $k^{2p+1}h^{2p}$
is sufficiently small, where $k$ is the wave number, $h$ is the mesh size, and
$E^{\rm PML}$ is the PML truncation error which is exponentially small. In
particular, the dependences of coefficients $C_j~(j=1,2)$ on the source $f$ are
improved. Numerical experiments are presented to validate the theoretical
findings, illustrating that the higher-order CIP-FEM can greatly reduce the
pollution errors.
