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In this paper, which is the second in a series of two, the preasymptotic 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 ≥ 1. By using a modified duality argument, preasymptotic error estimates are derived for both methods under the condition of (eqution presented), where k is the wave number, h is the mesh size, and C0 is a constant independent of k, h, p, and the penalty parameters. It is shown that the pollution errors of both methods in H1-norm are O(k2p+1h2p) if p = O(1) and are O(eqution presented) if the exact solution u ⋯ H 2(σ) which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here σ is a constant independent of k, h, p and the penalty parameters. Moreover, it is proved that the CIPFEM 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. © 2013 Society for Industrial and Applied Mathematics.

Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ Version