The dispersive and dissipative 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) (2002) 516]. 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.
- discrete dispersion relation
- high wave number
- discontinuous Galerkin approximation
- hp-finite element method
- computational physics