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Symplectic-mixed finite element approximation of linear acoustic wave equations

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Abstract

We apply mixed finite element approximations to the first-order form of the acoustic wave equation. The semidiscrete method exactly conserves the system energy. A fully discrete method employing the symplectic Euler time method in time exactly conserves a positive-definite pertubed energy functional that is equivalent to the actual energy under a CFL condition. In addition to proving optimal-order \(L^\infty (L^2)\) estimates, we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions.

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Acknowledgments

This was was supported by the National Science Foundation under Award Number 1117794, and the first author wishes to thank Prof. Roland Glowinski for valuable pointers to the relevant literature and Dr. Colin Cotter for helpful discussions regarding the Hamiltonian nature of the mixed form.

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Authors and Affiliations

  1. Department of Mathematics, Baylor University, One Bear Place #97328, PO Box 1042, Waco, TX, 76798-7328, USA

    Robert C. Kirby

  2. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA

    Thinh Tri Kieu

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  1. Robert C. Kirby
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  2. Thinh Tri Kieu
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Correspondence to Robert C. Kirby.

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Kirby, R.C., Kieu, T.T. Symplectic-mixed finite element approximation of linear acoustic wave equations. Numer. Math. 130, 257–291 (2015). https://doi.org/10.1007/s00211-014-0667-4

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  • DOI: https://doi.org/10.1007/s00211-014-0667-4

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