Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes

Mark Ainsworth, Hafiz Abdul Wajid, Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder)

Research output: Chapter in Book/Report/Conference proceedingChapter

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.
LanguageEnglish
Title of host publicationComputer Methods in Mathematics
Subtitle of host publicationAdvanced Structured Materials
PublisherSpringer
Pages3-17
Number of pages15
Volume1
Edition1
DOIs
Publication statusPublished - 2010

Publication series

NameComputer Methods in Mathematics: Advanced Structured Materials
PublisherSpringer

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wave equations
acoustics
grids
tensors
products

Keywords

  • acoustic wave equation
  • finite element analysis
  • spectral elements

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
Ainsworth, Mark ; Wajid, Hafiz Abdul ; Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder). / Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes. Computer Methods in Mathematics: Advanced Structured Materials. Vol. 1 1. ed. Springer, 2010. pp. 3-17 (Computer Methods in Mathematics: Advanced Structured Materials).
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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.",
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author = "Mark Ainsworth and Wajid, {Hafiz Abdul} and {Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder)}",
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Ainsworth, M, Wajid, HA & 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 edn, vol. 1, Computer Methods in Mathematics: Advanced Structured Materials, Springer, pp. 3-17. https://doi.org/10.1007/978-3-642-05241-5_1

Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes. / Ainsworth, Mark; Wajid, Hafiz Abdul; Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder).

Computer Methods in Mathematics: Advanced Structured Materials. Vol. 1 1. ed. Springer, 2010. p. 3-17 (Computer Methods in Mathematics: Advanced Structured Materials).

Research output: Chapter in Book/Report/Conference proceedingChapter

TY - CHAP

T1 - Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes

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N2 - 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.

AB - 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.

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KW - spectral elements

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Ainsworth M, Wajid HA, Support of MA by the Engineering and Physical Sciences Research Council under grant (Funder). 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. Springer. 2010. p. 3-17. (Computer Methods in Mathematics: Advanced Structured Materials). https://doi.org/10.1007/978-3-642-05241-5_1