Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods

Authors:
Mark Ainsworth
Affiliations:
Department of Mathematics, Strathclyde University, 26 Richmond Street, Livingstone Tower, Glasgow G1 1XH, Scotland, UK
Venue:
Journal of Computational Physics
Year:
2004

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Abstract

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.