$hp$-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we develop and analyze some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only mesh-dependent (or $h$-dependent) but also degree-dependent (or $p$-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, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed $hp$-discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number $k$, sub-optimal order (with respect to $h$ and $p$) error estimates in the broken $H^1$-norm and the $L^2$-norm are derived without any mesh constraint. The error estimates as well as the stability estimates are improved to optimal order under the mesh condition $k^3h^2p^{-2}\le C_0$ by utilizing these stability and error estimates and using a stability-error iterative procedure, where $C_0$ is some constant independent of $k$, $h$, $p$, and the penalty parameters. To overcome the difficulty caused by strong indefiniteness (and non-Hermitian nature) 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 [19, 20, 33],

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.