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

TL;DR

This paper extends the pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation to higher polynomial degrees, providing error estimates, stability results, and insights into pollution effects for high wave numbers.

Contribution

It develops error estimates and stability results for CIP-FEM and FEM with polynomial degree p ≥ 1, enhancing understanding of pollution errors in high-frequency Helmholtz problems.

Findings

Pollution errors are characterized as O(k^{2p+1}h^{2p}) for p=O(1).

CIP-FEM is shown to be stable for all k, h, p > 0 with positive imaginary penalty parameters.

Penalty parameters can be tuned to reduce pollution effects in high wave number regimes.

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.