# Dispersive and dissipative behavior of the spectral element method

Ainsworth, Mark and Wajid, Hafiz Abdul
(2009)
*Dispersive and dissipative behavior of the spectral element method.*
SIAM Journal on Numerical Analysis, 47 (5).
pp. 3910-3937.
ISSN 0036-1429

## Abstract

If the nodes for the spectral element method are chosen to be the Gauss-Legendre-Lobatto points and a Lagrange basis is used, then the resulting mass matrix is diagonal and the method is sometimes then described as the Gauss-point mass lumped finite element scheme. We study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh-size and the order of the scheme. We prove that (a) the Gauss-point mass lumped scheme (i.e., spectral element method) tends to exhibit phase lag whereas the (consistent) finite element scheme tends to exhibit phase lead; (b) the absolute accuracy of the spectral element scheme is $1/p$ times better than that of the finite element scheme despite the use of numerical integration; (c) when the order $p$, the mesh-size $h$, and the frequency of the wave $\omega$ satisfy $2p+1 \approx \omega h$ the true wave is essentially fully resolved. As a consequence, one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods: $\pi$ modes per wavelength are needed to resolve a wave.

Item type: | Article |
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ID code: | 28041 |

Keywords: | mass lumped scheme, numerical dispersion, spectral element method, Mathematics, Numerical Analysis |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Mrs Carolynne Westwood |

Date Deposited: | 13 Oct 2010 09:25 |

Last modified: | 09 May 2016 00:02 |

URI: | http://strathprints.strath.ac.uk/id/eprint/28041 |