Advertisement

Journal of Scientific Computing

, Volume 61, Issue 2, pp 424–453 | Cite as

Variational Space–Time Methods for the Wave Equation

  • Uwe Köcher
  • Markus Bause
Article

Abstract

In this work we present some new variational space–time discretisations for the scalar-valued acoustic wave equation as a prototype model for the vector-valued elastic wave equation. The second-order hyperbolic equation is rewritten as a first-order in time system of equations for the displacement and velocity field. For the discretisation in time we apply continuous Galerkin–Petrov and discontinuous Galerkin methods, and for the discretisation in space we apply the symmetric interior penalty discontinuous Galerkin method. The resulting algebraic system of equations exhibits a block structure. First, it is simplified by some calculations to a linear system for one of the variables and a vector update for the other variable. Using the block diagonal structure of the mass matrix from the discontinuous Galerkin discretisation in space, the reduced system can be condensed further such that the overall linear system can be solved efficiently. The convergence behaviour of the presented schemes is studied carefully by numerical experiments. Moreover, the performance and stability properties of the schemes are illustrated by a more sophisticated problem with complex wave propagation phenomena in heterogeneous media.

Keywords

Space–time Galerkin methods Variational time discretisation  Symmetric interior penalty discontinuous Galerkin method Higher order finite element methods 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables), “25.4, Integration”. Dover Books on Mathematics, New York (1972)Google Scholar
  2. 2.
    Ahmed, N., Matthies, G.: Numerical studies of Galerkin-type time-discretizations applied to transient convection–diffusion–reaction equations. World Acad. Sci. Eng. Technol. 66, 586–593 (2012)Google Scholar
  3. 3.
    Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27(1–3), 5–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aziz, A.K., Monk, P.: Continuous finite elements in space and time for the heat equation. Math. Comput. 52, 255–274 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bangerth, W., Burstedde, C., Heister, T., Kronbichler, M.: Algorithms and data structures for massively parallel generic adaptive finite element codes. ACM Trans. Math. Softw. (2011). doi: 10.1145/2049673.2049678
  7. 7.
    Bangerth, W., Geiger, M., Rannacher, R.: Adaptive Galerkin finite element methods for the wave equation. Comput. Methos Appl. Math. 10(1), 3–48 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bangerth, W., Heister, T., Kanschat, G.: deal. II differential equations analysis library. Technical reference (2013). http://www.dealii.org
  9. 9.
    Bause, M., Köcher, U.: Numerical simulation of elastic wave propagation in composite material. In: Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering, pp. 1–18 (2012)Google Scholar
  10. 10.
    Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. J. Sci. Comput. 31(3), 1985–2014 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Efendiev, Y., Hou, T.H.: Multiscale Finite Element Methods. Springer, New York (2009)zbMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grote, M.J., Schötzau, D.: Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. J. Sci. Comput. 40, 257–272 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Heroux, M., et al.: An Overview of Trilinos. Sandia National Laboratories, SAND2003-2927 (2003)Google Scholar
  16. 16.
    Hoppe, R.H.W., Kanschat, G., Warburton, T.: Convergence analysis of an adaptive interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 47(1), 534–550 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hussain, S., Schieweck, F., Turek, S.: Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer. Math. 19(1), 41–61 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hussain, S., Schieweck, F., Turek, S.: Higher order Galerkin time discretization for nonstationary incompressible flow. Numer. Math. Adv. Appl. 2011, 509–517 (2013)Google Scholar
  19. 19.
    Hussain, S., Schieweck, F., Turek, S.: A note on accurate and efficient higher order Galerkin time stepping schemes for nonstationary Stokes equations. Open Numer. Methods J. 4, 35–45 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kanzow, C.: Numerik linearer Gleichungssysteme, Direkte und iterative Verfahren. Springer, Berlin (2005)zbMATHGoogle Scholar
  21. 21.
    Lions, J.L., Magenes, E.: Problèmes aus limites non homogènes et applications, 1, 2, 3. Dunod, Paris (1968)Google Scholar
  22. 22.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  23. 23.
    Matthies, G., Schieweck, F.: Higher order variational time discretizations for nonlinear systems of ordinary differential equations. Preprint no. 23/2011, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg (2011)Google Scholar
  24. 24.
    Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  25. 25.
    Schieweck, F.: A-stable discontinuous Galerkin–Petrov time discretization of higher order. J. Numer. Math. 18(1), 25–57 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Thomeé, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)zbMATHGoogle Scholar
  27. 27.
    Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Helmut Schmidt UniversityHamburgGermany

Personalised recommendations