Skip to main content
SpringerLink
Log in

Convergence Analysis of Hybrid High-Order Methods for the Wave Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We prove error estimates for the wave equation semi-discretized in space by the hybrid high-order (HHO) method. These estimates lead to optimal convergence rates for smooth solutions. We consider first the second-order formulation in time, for which we establish \({\varvec{H}}^1\) and \({\varvec{L}}^2\)-error estimates, and then the first-order formulation, for which we establish \({\varvec{H}}^1\)-error estimates. For both formulations, the space semi-discrete HHO scheme has close links with hybridizable discontinuous Galerkin schemes from the literature. Numerical experiments using either the Newmark scheme or diagonally-implicit Runge–Kutta schemes for the time discretization illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of elastic waves in heterogeneous media.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availibility

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Abbas, M., Ern, A., Pignet, N.: Hybrid high-order methods for finite deformations of hyperelastic materials. Comput. Mech. 62(4), 909–928 (2018)

    Article  MathSciNet  Google Scholar 

  2. Abbas, M., Ern, A., Pignet, N.: A hybrid high-order method for incremental associative plasticity with small deformations. Comput. Methods Appl. Mech. Eng. 346, 891–912 (2019)

    Article  MathSciNet  Google Scholar 

  3. Baker, G.A.: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13(4), 564–576 (1976)

    Article  MathSciNet  Google Scholar 

  4. Boillot, L.: Contributions to the mathematical modeling and to the parallel algorithmic for the optimization of an elastic wave propagator in anisotropic media. Ph.D. thesis, Université de Pau et des Pays de l’Adour, December (2014)

  5. Botti, M., Di Pietro, D.A., Sochala, P.: A hybrid high-order method for nonlinear elasticity. SIAM J. Numer. Anal. 55(6), 2687–2717 (2017)

    Article  MathSciNet  Google Scholar 

  6. Burman, E., Duran, O., Ern, A.: Hybrid high-order methods for the acoustic wave equation in the time domain. Commun. Appl. Math. Comput. (to appear). Available at https://hal.archives-ouvertes.fr/hal-02922702 (2020)

  7. Chou, C.-S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014)

    Article  MathSciNet  Google Scholar 

  8. Chouly, F., Ern, A., Pignet, N.: A hybrid high-order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity. SIAM J. Sci. Comput. 42(4), A2300–A2324 (2020)

    Article  MathSciNet  Google Scholar 

  9. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44(5), 2131–2158 (2006)

    Article  MathSciNet  Google Scholar 

  10. Cicuttin, M., Di Pietro, D.A., Ern, A.: Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming. J. Comput. Appl. Math. 344, 852–874 (2018)

    Article  MathSciNet  Google Scholar 

  11. Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model Numer. Anal. 50(3), 635–650 (2016)

    Article  MathSciNet  Google Scholar 

  12. Cockburn, B., Fu, Z., Hungria, A., Ji, L., Sánchez, M.A., Sayas, F.-J.: Stormer-Numerov HDG methods for acoustic waves. J. Sci. Comput. 75(2), 597–624 (2018)

    Article  MathSciNet  Google Scholar 

  13. Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79(271), 1351–1367 (2010)

    Article  MathSciNet  Google Scholar 

  14. Cockburn, B., Quenneville-Bélair, V.: Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation. Math. Comput. 83(285), 65–85 (2014)

    Article  MathSciNet  Google Scholar 

  15. Cohen, G.C.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2002)

    Book  Google Scholar 

  16. Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  MathSciNet  Google Scholar 

  17. Di Pietro, D.A., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14(4), 461–472 (2014)

    Article  MathSciNet  Google Scholar 

  18. Dupont, T.: \(L^{2}\)-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10, 880–889 (1973)

    Article  MathSciNet  Google Scholar 

  19. Falk, R.S., Richter, G.R.: Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36(3), 935–952 (1999)

    Article  MathSciNet  Google Scholar 

  20. Griesmaier, R., Monk, P.: Discretization of the wave equation using continuous elements in time and a hybridizable discontinuous Galerkin method in space. J. Sci. Comput. 58(2), 472–498 (2014)

    Article  MathSciNet  Google Scholar 

  21. Grote, M.J., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006)

    Article  MathSciNet  Google Scholar 

  22. Lehrenfeld, C.: Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Ph.D. thesis, Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen (2010)

  23. Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)

    Article  MathSciNet  Google Scholar 

  24. Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, vols. I, II. Springer, New York. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band, pp. 181–182 (1972)

  25. Monk, P., Richter, G.R.: A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media. J. Sci. Comput. 22(23), 443–477 (2005)

    Article  MathSciNet  Google Scholar 

  26. Nguyen, N.C., Peraire, J., Cockburn, B.: High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230(10), 3695–3718 (2011)

    Article  MathSciNet  Google Scholar 

  27. Sánchez, M.A., Ciuca, C., Nguyen, N.C., Peraire, J., Cockburn, B.: Symplectic Hamiltonian HDG methods for wave propagation phenomena. J. Comput. Phys. 350, 951–973 (2017)

    Article  MathSciNet  Google Scholar 

  28. Stanglmeier, M., Nguyen, N.C., Peraire, J., Cockburn, B.: An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Comput. Methods Appl. Mech. Eng. 300, 748–769 (2016)

    Article  MathSciNet  Google Scholar 

  29. Wheeler, M.F.: A priori \(L_{2}\) error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10, 723–759 (1973)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank L. Guillot (CEA/DAM) and F. Drui and O. Jamond (CEA/DEN) for insightful discussions and CEA/DAM for partial financial support. EB was partially supported by the EPSRC Grants EP/P01576X/1 and EP/P012434/1.

Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, London, WC1E 6BT, UK

    Erik Burman

  2. CERMICS, Ecole des Ponts, 77455, Marne la Vallée cedex 2, France

    Omar Duran, Alexandre Ern & Morgane Steins

  3. INRIA Paris, 75589, Paris, France

    Omar Duran, Alexandre Ern & Morgane Steins

  4. DEN-Service d’étude mécanique et thermiques (SEMT), CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France

    Morgane Steins

Authors
  1. Erik Burman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Duran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandre Ern
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Morgane Steins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre Ern.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burman, E., Duran, O., Ern, A. et al. Convergence Analysis of Hybrid High-Order Methods for the Wave Equation. J Sci Comput 87, 91 (2021). https://doi.org/10.1007/s10915-021-01492-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01492-1

Keywords

Mathematics Subject Classification

Search

Navigation