PREASYMPTOTIC ERROR ANALYSIS OF CIP-FEM AND FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER. PART II: hp VERSION

• Published in: SIAM journal on numerical analysis Vol. 51; no. 3; pp. 1828 - 1852
• Main Authors: ZHU, LINGXUE, WU, HAIJUN
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Approximation; Error analysis; Estimates; Finite element method; Helmholtz equations; Mathematical analysis; Mathematical models; Pollution; Pollution abatement; Wavelengths
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/120874643

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 ≥ 1. By using a modified duality argument, preasymptotic error estimates are derived for both methods under the condition of $\frac{{kh}}{p} \leqslant Co{(\frac{p}{k})^{\frac{1}{{p + 1}}}$ , where k is the wave number, h is the mesh size, and Co 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(k² p +¹h² p ) if p = O(1) and are O $(\frac{k}{{{p^2}}}{(\frac{{kh}}{{\sigma p}})^{2p}})$ if the exact solution u ϵ H²(Ω) which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here σ is a constant independent of k, h, p and the penalty parameters. Moreover, it is proved that the CIPFEM 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.