Advertisement

hp-Version Discontinuous Galerkin Approximations of the Elastodynamics Equation

  • Paola F. AntoniettiEmail author
  • Alberto Ferroni
  • Ilario Mazzieri
  • Alfio Quarteroni
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

In this paper we extend the results contained in Antonietti et al. (J Sci Comput 68(1):143-170, 2016) and consider the problem of approximating the elastodynamics equation by means of hp-version discontinuous Galerkin methods. For the resulting semi-discretized schemes we derive stability bounds as well as hp error estimates in the energy and L 2-norms. Our theoretical estimates are verified through three dimensional numerical experiments.

Notes

Acknowledgements

Paola F. Antonietti and Ilario Mazzieri have been partially supported by the research grant no. 2015-0182 funded by Fondazione Cariplo and Regione Lombardia. Part of this work has been completed while Paola F. Antonietti was visiting the Institut Henri Poincaré (IHP), Paris. She thanks the Institute for the kind hospitality.

References

  1. 1.
    M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Ainsworth, R. Rankin, Technical note: a note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes. Numer. Methods Partial Differential Equations 28(3), 1099–1104 (2012)CrossRefMathSciNetGoogle Scholar
  3. 3.
    P.F. Antonietti, I. Mazzieri, A. Quarteroni, F. Rapetti, Non-conforming high order approximations of the elastodynamics equation. Comput. Methods Appl. Mech. Eng. 209–212, 212–238 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    P.F. Antonietti, C. Marcati, I. Mazzieri, A. Quarteroni, High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation. Numer. Algorithms 71(1), 181–206 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    P.F. Antonietti, B. Ayuso de Dios, I. Mazzieri, A. Quarteroni, Stability analysis of discontinuous Galerkin approximations to the elastodynamics problem. J. Sci. Comput. 68(1), 143–170 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    P.F. Antonietti, A. Ferroni, I. Mazzieri, R. Paolucci, A. Quarteroni, C. Smerzini, M. Stupazzini, Numerical modeling of seismic waves by discontinuous Spectral Element methods. MOX Report 9/2017 (Submitted, 2017)Google Scholar
  7. 7.
    D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    I. Babuŝka, M. Suri, The hp version of the finite element method with quasiuniform meshes. Math. Model. Numer. Anal. 21(2), 199–238 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S.C. Brenner, Korn’s inequalities for piecewise H 1 vector fields. Math. Comput. 73(247), 1067–1087 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    J.D. De Basabe, M.K. Sen, M.F. Wheeler, The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175(1), 83–93 (2008)CrossRefGoogle Scholar
  12. 12.
    M. Dumbser, M. Käser, An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes - ii. the three-dimensional isotropic case. Geophys. J. Int. 167(1), 319–336 (2006)Google Scholar
  13. 13.
    T. Dupont, L 2-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10, 880–889 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    E. Faccioli, F. Maggio, R. Paolucci, A. Quarteroni, 2d and 3d elastic wave propagation by a pseudo-spectral domain decomposition method. J. Seimol. 1(3), 237–251 (1997)CrossRefGoogle Scholar
  15. 15.
    A. Ferroni, P.F. Antonietti, I. Mazzieri, A. Quarteroni, Dispersion-dissipation analysis of 3D continuous and discontinuous Spectral Element methods for the elastodynamics equation. MOX Report 18/2016 (Submitted)Google Scholar
  16. 16.
    E.H. Georgoulis, E. Hall, P. Houston, Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput. 30(1), 246–271 (2007/2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    E.H. Georgoulis, E. Süli, Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA J. Numer. Anal. 25(1), 205–220 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    N.A. Haskell, The dispersion of surface waves on multi-layered media. Bull. Seismol. Soc. Am. 43, 17–34 (1953)MathSciNetGoogle Scholar
  19. 19.
    P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    D. Komatitsch, J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)CrossRefGoogle Scholar
  21. 21.
    D. Komatitsch, J. Tromp, Spectral-element simulations of global seismic wave propagation - i. validation. Geophys. J. Int. 149(2), 390–412 (2002)Google Scholar
  22. 22.
    D. Komatitsch, J.-P. Vilotte, R. Vai, J. Castillo-Covarrubias, F. Snchez-Sesma, The spectral element method for elastic wave equations - application to 2-d and 3-d seismic problems. Int. J. Numer. Meth. Eng. 45(9), 1139–1164 (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    D. Komatitsch, J. Ritsema, J. Tromp, Geophysics: the spectral-element method, beowulf computing, and global seismology. Science 298(5599), 1737–1742 (2002)CrossRefGoogle Scholar
  24. 24.
    M. Kser, M. Dumbser, An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - i. the two-dimensional isotropic case with external source terms. Geophys. J. Int. 166(2), 855–877 (2006)Google Scholar
  25. 25.
    D.J.P. Lahaye, F. Maggio, A. Quarteroni, Hybrid finite element–spectral element approximation of wave propagation problems. East-West J. Numer. Math. 5(4), 265–289 (1997)zbMATHMathSciNetGoogle Scholar
  26. 26.
    I. Mazzieri, M. Stupazzini, R. Guidotti, C. Smerzini, Speed: spectral elements in elastodynamics with discontinuous Galerkin: a non-conforming approach for 3d multi-scale problems. Int. J. Numer. Meth. Eng. 95(12), 991–1010 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    I. Mazzieri, M. Stupazzini, R. Guidotti, C. Smerzini, Speed: spectral elements in elastodynamics with discontinuous Galerkin: a non-conforming approach for 3D multi-scale problems. Int. J. Numer. Methods Eng. 95(12), 991–1010 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    A.T. Patera, Spectral methods for spatially evolving hydrodynamic flows, in Spectral Methods for Partial Differential Equations (Hampton, VA, 1982) (SIAM, Philadelphia, 1984), pp. 239–256Google Scholar
  29. 29.
    I. Perugia, D. Schötzau, An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17(1–4), 561–571 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    A. Quarteroni, Numerical models for differential problems, vol. 8, 2nd edn. MS&A. Modeling, Simulation and Applications. (Springer, Milan, 2014). Translated from the fifth (2012) Italian edition by Silvia QuarteroniGoogle Scholar
  31. 31.
    B. Rivière, M.F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, in Current Trends in Scientific Computing (Xi’an, 2002), vol. 329, Contemporary Mathematics (American Mathematical Society, Providence, 2003), pp. 271–282Google Scholar
  32. 32.
    B. Rivière, S. Shaw, M.F. Wheeler, J.R. Whiteman, Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95(2), 347–376 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    B. Rivière, S. Shaw, J.R. Whiteman, Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems. Numer. Methods Partial Differential Equations 23(5), 1149–1166 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    C. Schwab, p- and hp-Finite Element Methods. Numerical Mathematics and Scientific Computation (The Clarendon Press, Oxford University Press, New York, 1998). Theory and applications in solid and fluid mechanicsGoogle Scholar
  35. 35.
    G. Seriani, E. Priolo, A. Pregarz, Modelling waves in anisotropic media by a spectral element method, in Mathematical and Numerical Aspects of Wave Propagation (Mandelieu-La Napoule, 1995) (SIAM, Philadelphia, 1995), pp. 289–298zbMATHGoogle Scholar
  36. 36.
    C. Smerzini, R. Paolucci, M. Stupazzini, Experimental and numerical results on earthquake-induced rotational ground motions. J. Earthq. Eng. 13(Suppl. 1), 66–82 (2009)CrossRefGoogle Scholar
  37. 37.
    B. Stamm, T.P. Wihler, hp-optimal discontinuous Galerkin methods for linear elliptic problems. Math. Comput. 79(272), 2117–2133 (2010)Google Scholar
  38. 38.
    R. Stenberg, Mortaring by a method of J. A. Nitsche, in Computational Mechanics (Buenos Aires, 1998) (Centro Internac. Métodos Numér. Ing., Barcelona, 1998)Google Scholar
  39. 39.
    M. Stupazzini, R. Paolucci, H. Igel, Near-fault earthquake ground-motion simulation in the grenoble valley by a high-performance spectral element code. Bull. Seismol. Soc. Am. 99(1), 286–301 (2009)CrossRefGoogle Scholar
  40. 40.
    W.T. Thomson, Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21, 89–93 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Paola F. Antonietti
    • 1
    Email author
  • Alberto Ferroni
    • 1
  • Ilario Mazzieri
    • 1
  • Alfio Quarteroni
    • 2
  1. 1.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.CMCS, Ecole Polytechnique Federale de Lausanne (EPFL)LausanneSwitzerland

Personalised recommendations