Dispersive Behaviour of High Order Discontinuous Galerkin Finite Element Methods

The dispersive properties of hp version discontinuous Galerkin finite element approximation are studied in three different limits. For the small wave-number limit hk → 0, we show the discontinuous Galerkin gives a higher order of accuracy than the standard Galerkin procedure, thereby confirming the conjectures of Hu and Atkins (J. Comput. Phys., 182(2):516– 545, 2002 ). If the mesh is fixed and the order p is increased, it is shown that the dissipation and dispersion errors decay at a super-exponential rate when the order p is much larger than hk. Finally, if the order is chosen so that 2p + 1 ≈ κhk for some fixed constant κ > 1, then it is shown that an exponential rate of decay is obtained.