$hp$-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order $p$. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed $hp$-discontinuous Galerkin methods are absolutely stable (hence, well-posed). For each fixed wave number $k$, sub-optimal order error estimates in the broken $H^1$-norm and the $L^2$-norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition $k^3h^2p^{-1}\le C_0$ by utilizing these stability and error estimates and using a stability-error iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}, which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size $h$, the polynomial degree $p$, the wave number $k$, as well as all the penalty parameters for the numerical solutions.