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

Abstract

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 $p \ge 1$. By using a modified duality argument, preasymptotic error estimates are derived for both methods under the condition of $\frac{kh}{p}\le C_0(\frac{p}{k})^{\frac{1}{p+1}}$, 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 $O(\frac{k}{p^2}(\frac{kh}{\sigma p})^{2p})$ if the exact solution $u\in H^2(\Omega)$ which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here $\sigma$ is a constant independent of $k, h, p$ and the penalty par...