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A preasymptotic error analysis of the finite element method (FEM) and somecontinuous interior penalty finite element method (CIP-FEM) for Helmholtzequation in two and three dimensions is proposed. $H^1$- and $L^2$- errorestimates with explicit dependence on the wave number $k$ are derived. Inparticular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, thenthe 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 obtainedby existent dispersion analyses on Cartesian grids, where $h$ is the mesh size,$p$ is the order of the approximation space and is fixed. The CIP-FEM extendsthe classical one by adding more penalty terms on jumps of higher (up to $p$-thorder) normal derivatives in order to reduce efficiently the pollution errorsof higher order methods. Numerical tests are provided to verify the theoreticalfindings and to illustrate great capability of the CIP-FEM in reducing thepollution effect.

Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number