Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. II: \(hp\) version

DOI10.1137/120874643zbMATH Open1416.65468arXiv1204.5061OpenAlexW2004475615MaRDI QIDQ2845614

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Publication date: 2 September 2013

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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 $pge1$. By using a modified duality argument, pre-asymptotic error estimates are derived for both methods under the condition of $$, where $k$ is the wave number, $h$ is the mesh size, and $C_0$ is a constant independent of $k,h,p$, and the penalty parameters. It is shown that the pollution errors of both methods in $H^1$-norm are $O(k^{2p+1}{h}^{2p})$ if $p=O(1)$ and are $$ if the exact solution $uin{H}^2(Om)$ 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.

Full work available at URL: https://arxiv.org/abs/1204.5061

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