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

Abstract

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.