  In this paper, which is part II in a series of two, the pre-asymptotic error
analysis of the continuous interior penalty finite element method (CIP-FEM) and
the FEM for the Helmholtz equation in two and three dimensions is continued.
While part I contained results on the linear CIP-FEM and FEM, the present part
deals with approximation spaces of order $p \ge 1$. By using a modified duality
argument, pre-asymptotic error estimates are derived for both methods under the
condition of $\frac{kh}{p}\le C_0\big(\frac{p}{k}\big)^{\frac{1}{p+1}}$, where
$k$ is the wave number, $h$ is the mesh size, and $C_0$ is a constant
independent of $k, h, p$, and the penalty parameters. It is shown that the
pollution errors of both methods in $H^1$-norm are $O(k^{2p+1}h^{2p})$ if
$p=O(1)$ and are $O\Big(\frac{k}{p^2}\big(\frac{kh}{\sigma p}\big)^{2p}\Big)$
if the exact solution $u\in H^2(\Om)$ which coincide with existent dispersion
analyses for the FEM on Cartesian grids. Here $\si$ is a constant independent
of $k, h, p$, and the penalty parameters. Moreover, it is proved that the
CIP-FEM is stable for any $k, h, p>0$ and penalty parameters with positive
imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM
compared to the FEM, the penalty parameters may be tuned to reduce the
pollution effects.
