### 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.

Original language | English |
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Pages (from-to) | 106-130 |

Number of pages | 24 |

Journal | Journal of Computational Physics |

Volume | 198 |

Issue number | 1 |

DOIs | |

Publication status | Published - 20 Jul 2004 |

### Keywords

- discrete dispersion relation
- high wave number
- discontinuous Galerkin approximation
- hp-finite element method
- computational physics

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## Cite this

Ainsworth, M. (2004). Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods.

*Journal of Computational Physics*,*198*(1), 106-130. https://doi.org/10.1016/j.jcp.2004.01.004