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Dispersion analysis of triangle-based spectral element methods for elastic wave propagation

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Abstract

We study the numerical dispersion/dissipation of Triangle-based Spectral Element Methods (TSEM) of order N ≥ 1 when coupled with the Leap-Frog (LF) finite difference scheme to simulate the elastic wave propagation over a structured triangulation of the 2D physical domain. The analysis relies on the discrete eigenvalue problem resulting from the approximation of the dispersion relation. First, we present semi-discrete dispersion graphs by varying the approximation polynomial degree and the number of discrete points per wavelength. The fully-discrete ones are then obtained by varying also the time step. Numerical results for the TSEM, resp. TSEM-LF, are compared with those of the classical Quadrangle-based Spectral Element Method (QSEM), resp. QSEM-LF.

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References

  1. Briani, M., Sommariva, A., Vianello, M.: Computing Fekete and Lebesgue points: simplex, square, disk. J. Comput. Appl. Math. 236, 2477–2486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernardi, C., Maday, Y.: Some spectral approximations of monodimensional fourth order problems. In: Nevai, P., Pinkus, A. (eds.) Progress in Approximation Theory (1991)

  3. Capdeville, Y., Chaljub, E., Vilotte, J.P., Montagner, J.P.: Coupling the spectral element method with a modal solution for elastic wave propagation in realistic 3D global earth models. Geophys. J. Int. 152(1), 34–68 (2003)

    Article  Google Scholar 

  4. Carcione, J.M., Kosloff, D., Behle, A., Seriani, G.: A spectral scheme for wave propagation simulation in 3-D elasticanisotropic media. Geophysics 57, 1593 (1992)

    Google Scholar 

  5. Chaljub, E., Capdeville, Y., Vilotte, J.P.: Solving elastodynamics in a fluidsolid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids. J. Comput. Phys. 187(2), 457–491 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Basabe, J.D., Sen, M.K.: Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations. Geophysics 72/6, 81–95 (2007)

    Google Scholar 

  7. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Faccioli, E., Maggio, F., Quarteroni, A., Tagliani, A.: Spectral-domain decomposition methods for the solution of acoustic and elastic wave equation. Geophysics 61, 1160–1174 (1996)

    Google Scholar 

  9. Faccioli, E., Maggio, F., Paolucci, R., Quarteroni, A.: 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method. J. Seismol. 1, 237–251 (1997)

    Article  Google Scholar 

  10. Hughes, T.J.R.: The Finite Element Method, 2nd edn. Dover Publications, Mineola, NY (2000)

    MATH  Google Scholar 

  11. Giraldo, F.X., Warburton, T.: A nodal triangle-based spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 207(1), 129–150 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, London (2005)

    Book  Google Scholar 

  13. Komatitsch, D., Martin, R., Tromp, J., Taylor, M.A., Wingate, B.A.: Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles. J. Comput. Acoust. 9, 703–718 (2001)

    Article  Google Scholar 

  14. Komatitsch, D., Tromp, J.: Introduction to the spectral-element method for 3-D seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)

    Article  Google Scholar 

  15. Komatitsch, D., Liu, Q., Tromp, J., Sussa, P., Stidham, C., Shaw, J.H.: Simulations of ground motion in the Los Angeles basin based upon the spectral-element method. Bull. Seismol. Soc. Am. 94(1), 187–206 (2004)

    Article  Google Scholar 

  16. Komatitsch, D., Barnes, C., Tromp, J.: Wave propagation near a fluid-solid interface: a spectral element approach. Geophysics 65, 623–631 (2000)

    Google Scholar 

  17. Komatitsch, D., Vilotte, J.P., Vai, R., Castillo-Covarrubias, J.M., Sanchez-Sesma, F.J.: Spectral element approximation of elastic waves equations: application to 2-D and 3-D seismic problems. Int. J. Numer. Methods Eng. 45, 1139–1164 (1999)

    Article  MATH  Google Scholar 

  18. Komatitsch, D., Vilotte, J.P.: The spectral element method: an efficient tool to simulate the seismic response of 2-D and 3-D geological structures. Bull. Seismol. Soc. Am. 88, 368–392 (1998)

    Google Scholar 

  19. Koorwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R.A. (ed.) Theory and Applications of Special Functions, pp. 435–495. Academic Press (1975)

  20. Maday, Y., Patera, A.T.: Spectral element methods for the incompressible Navier–Stokes equations. In: State of the Art Surveys in Computational Mechanics, pp. 71–143. ASME (1989)

  21. Mercerat, E.D., Vilotte, J.P., Sánchez-Sesma, F.J.: Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids. Geophys. J. Int. 166/2, 679–698 (2006)

    Article  Google Scholar 

  22. Mitchell, A., Griffiths, D.: The Finite Difference Method in Partial Differential Equations. John Wiley and Sons (1980)

  23. Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: comparisons and recent advances. J. Sci. Comput. 27, 377–387 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: which interpolation points? Numer. Algorithms 55/2–3, 349–366 (2010)

    Article  MathSciNet  Google Scholar 

  25. Patera, A.T.: A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)

    Article  MATH  Google Scholar 

  26. Raviart, P.-A., Thomas, J.-M.: Introduction à l’analyse Numérique des Équations aux Dérivées Partielles. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983)

  27. Seriani, G., Oliveira, S.P.: Dispersion analysis of spectral element methods for elastic wave propagation. Wave Motion 45, 729–744 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Taylor, M.A., Wingate, B.A., Vincent, R.E.: An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal. 38(5), 1707–1720 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Warburton, T., Pavarino, L., Hesthaven, J.S.: A pseudo-spectral scheme for the incompressible Navier–Stokes equations using unstructured nodal elements. J. Comput. Phys. 164, 1–21 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, MOX-Modelling and Scientific Computing, P.za Leonardo da Vinci, 32, 20133, Milano, Italy

    Ilario Mazzieri

  2. LJAD-Laboratoire de Mathématiques “J.A. Dieudonné”, UMR 7351 UNSA/CNRS, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice, Cedex 02, France

    Francesca Rapetti

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  1. Ilario Mazzieri
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  2. Francesca Rapetti
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Correspondence to Francesca Rapetti.

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Mazzieri, I., Rapetti, F. Dispersion analysis of triangle-based spectral element methods for elastic wave propagation. Numer Algor 60, 631–650 (2012). https://doi.org/10.1007/s11075-012-9592-8

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