Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number

A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is proposed. $H^1$- and $L^2$-error estimates with explicit dependence on the wave number $k$ are derived. In particular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h^{2p})$, which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where $h$ is the mesh size, and $p$ is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to $p$th order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.