The continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz equation in two and three dimensions are considered.  By using a modified duality argument, pre-asymptotic error estimates are derived for both methods.  It is shown that the pollution errors of both methods in H^1-norm are O(k^{2p+1}h^{2p}).  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.