Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we shall consider to improve the pre-asymptotic stability and error estimates of some hp-interior penalty discontinuous Galerkin (hp-IPDG) methods for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in Feng and Wu (2011). The proposed hp-IPDG methods are defined using a sesquilinear form which is not only mesh-dependent (or h-dependent) but also degree-dependent (or p-dependent). By using a modified duality argument given in Zhu and Wu (2013), pre-asymptotic error estimates are improved for the proposed hp-IPDG methods under the condition of khp≤C0(pk)1p+1 in this paper, where C0 is some constant independent of k,h,p, and the penalty parameters. It is shown that the pollution error of the method in the broken H1-norm is O(k2p+1h2p) if p=O(1) which coincides with existent dispersion analyses for the DG method on Cartesian grids. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the IPDG method in reducing the pollution effect.