Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

A preasymptotic error analysis of the interior penalty discontinuous Galerkin (IPDG) method of high order for Helmholtz equation with the first order absorbing boundary condition in two and three dimensions is proposed. We derive the H-1- and L-2-error estimates with explicit dependence on the wave number k. In particular, it is shown that if k(kh)(2p) is sufficiently small, then the pollution errors of IPDG method in H-1-norm are bounded by O(k(kh)(2p)), which coincides with the phase error of the finite element method obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, p is the order of the approximation space and is fixed. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the symmetric IPDG method in reducing the pollution effect.