2012/04/23 by Lingxue Zhu, Zhu, Lingxue, Haijun Wu +1 · 1 citation
Engineering · #65N12 #65N15 #65N30 #78A40 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1204.5061
openalex publication_date 2012/04/23 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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 ≥ 1. By using a modified duality argument, pre-asymptotic error estimates are derived for both methods under the condition of (kh)/(p)≤ C0((p)/(k))(1)/(p+1), 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((k)/(p2)((kh)/(σp))2p) if the exact solution u∈ H2(\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.