### Abstract

We study the dispersive properties of the acoustic wave equation for finite element, spectral element and optimally blended schemes using tensor product elements defined on rectangular grid in d-dimensions. We prove and give analytical expressions for the discrete dispersion relations for the above mentioned schemes. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter is p/(p + 1) for all p ∈ ℕ and for any number of spatial dimensions; (c) the optimal scheme guarantees two additional orders of accuracy compared with both finite and spectral element schemes; and (d) the absolute accuracy of the optimally blended scheme is and times better than that of the pure finite and spectral element schemes respectively.

Original language | English |
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Title of host publication | Computer Methods in Mathematics |

Subtitle of host publication | Advanced Structured Materials |

Publisher | Springer |

Pages | 3-17 |

Number of pages | 15 |

Volume | 1 |

Edition | 1 |

DOIs | |

Publication status | Published - 2010 |

### Publication series

Name | Computer Methods in Mathematics: Advanced Structured Materials |
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Publisher | Springer |

### Keywords

- acoustic wave equation
- finite element analysis
- spectral elements

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## Cite this

Ainsworth, M., Wajid, H. A., & Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder) (2010). Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes. In

*Computer Methods in Mathematics: Advanced Structured Materials*(1 ed., Vol. 1, pp. 3-17). (Computer Methods in Mathematics: Advanced Structured Materials). Springer. https://doi.org/10.1007/978-3-642-05241-5_1