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Some Complex Models

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Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

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Abstract

This chapter presents three more complex equations with specific properties: The linearized Euler equations (LEE) which model acoustics in flow and contains a convective term, the Cauchy-Poisson problem which models gravity waves and whose evolution equation is on a boundary and two models of wave propagation in thin plates which are dispersive equations.

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Notes

  1. 1.

    We recall that the acoustics equation is their 0th order approximation.

  2. 2.

    Which was proven (in an unpublished paper) to be equivalent to LEE.

  3. 3.

    For tetrahedra or triangles, \(Q_r\) is replaced by \(P_r\).

  4. 4.

    A “hypermixed”, i.e. a mixed formulation in two steps, was constructed, but it was not stable.

  5. 5.

    This approach is a common work of G. Cohen, A. Hüppe, S. Imperiale and M. Kaltenbacher. An article entitled “Construction and analysis of an adapted spectral finite element method to convective acoustic equations” has been accepted by the Communications in Computational Physics Journal and should be published in 2016.

  6. 6.

    p being in \(H^1(\varOmega )\), \({\underline{u}}_0\) can no longer be piecewise constant.

  7. 7.

    Although, from a chronological point of view, the Reissner–Mindlin model is a generalization of the Kirchhoff–Love model.

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Authors and Affiliations

  1. Inria, Saclay, France

    Gary Cohen

  2. The French Aerospace Lab, ONERA, Toulouse, France

    Sébastien Pernet

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  1. Gary Cohen
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Correspondence to Gary Cohen .

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Cohen, G., Pernet, S. (2017). Some Complex Models. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_8

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  • DOI: https://doi.org/10.1007/978-94-017-7761-2_8

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  • Print ISBN: 978-94-017-7759-9

  • Online ISBN: 978-94-017-7761-2

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