PREASYMPTOTIC ERROR ANALYSIS OF CIP-FEM AND FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER. PART II: hp VERSION - ZHU, LINGXUE; WU, HAIJUN (SIAM journal on numerical analysis, 2013)
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 $(\frac{k}{{{p^2}}}{(\frac{{kh}}{{σ p}})^{2p}})$ 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.