Numerical Algorithms

, Volume 60, Issue 4, pp 631–650

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

• Ilario Mazzieri
• Francesca Rapetti
Original Paper

## 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.

## Keywords

Elastic wave equation Spectral element methods Triangular grids Dispersion/dissipation analysis

## References

1. 1.
Briani, M., Sommariva, A., Vianello, M.: Computing Fekete and Lebesgue points: simplex, square, disk. J. Comput. Appl. Math. 236, 2477–2486 (2012)
2. 2.
Bernardi, C., Maday, Y.: Some spectral approximations of monodimensional fourth order problems. In: Nevai, P., Pinkus, A. (eds.) Progress in Approximation Theory (1991)Google Scholar
3. 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)
4. 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. 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)
6. 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. 7.
Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6, 345–390 (1991)
8. 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. 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)
10. 10.
Hughes, T.J.R.: The Finite Element Method, 2nd edn. Dover Publications, Mineola, NY (2000)
11. 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)
12. 12.
Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, London (2005)
13. 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)
14. 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)
15. 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)
16. 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. 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)
18. 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. 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)Google Scholar
20. 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)Google Scholar
21. 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)
22. 22.
Mitchell, A., Griffiths, D.: The Finite Difference Method in Partial Differential Equations. John Wiley and Sons (1980)Google Scholar
23. 23.
Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: comparisons and recent advances. J. Sci. Comput. 27, 377–387 (2006)
24. 24.
Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: which interpolation points? Numer. Algorithms 55/2–3, 349–366 (2010)
25. 25.
Patera, A.T.: A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)
26. 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)Google Scholar
27. 27.
Seriani, G., Oliveira, S.P.: Dispersion analysis of spectral element methods for elastic wave propagation. Wave Motion 45, 729–744 (2008)
28. 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)
29. 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)