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Journal of Scientific Computing

, Volume 72, Issue 3, pp 1196–1213 | Cite as

Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes

  • Daniel Peterseim
  • Mira Schedensack
Article

Abstract

The Courant–Friedrichs–Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

Keywords

CFL condition Hyperbolic equation Finite element method Adaptive mesh refinement 

Mathematics Subject Classification

65M12 65M60 35L05 

Notes

Acknowledgements

The authors would like to thank Andreas Longva for pointing out that mass lumping indeed works. Parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics (Bonn).

References

  1. 1.
    Abdulle, A., Henning, P.: Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comput. 86(304), 549–587 (2017)Google Scholar
  2. 2.
    Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36(153), 35–51 (1981). doi: 10.2307/2007724 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65(215), 897–921 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Carstensen, C., Gallistl, D., Schedensack, M.: \(L^2\) best-approximation of the elastic stress in the Arnold–Winther FEM. IMA J. Numer. Anal. 36(3), 1096–1119 (2016). doi: 10.1093/imanum/drv051 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Christiansen, S.H.: Foundations of finite element methods for wave equations of Maxwell type. In: Quak, E., Soomere, T. (eds.) Applied Wave Mathematics, pp. 335–393. Springer, Berlin (2009)CrossRefGoogle Scholar
  7. 7.
    Ciarlet Jr., P., He, J.: The singular complement method for 2d scalar problems. C. R. Math. Acad. Sci. Paris 336(4), 353–358 (2003). doi: 10.1016/S1631-073X(03)00030-X MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928). doi: 10.1007/BF01448839 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-22980-0 Google Scholar
  10. 10.
    Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31(3), 1985–2014 (2009). doi: 10.1137/070709414 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diaz, J., Grote, M.J.: Multi-level explicit local time-stepping methods for second-order wave equations. Comput. Methods Appl. Mech. Eng. 291, 240–265 (2015). doi: 10.1016/j.cma.2015.03.027 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Elfverson, D., Georgoulis, E.H., Målqvist, A., Peterseim, D.: Convergence of a discontinuous Galerkin multiscale method. SIAM J. Numer. Anal. 51(6), 3351–3372 (2013). doi: 10.1137/120900113 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ern, A., Guermond, J.L.: Finite element quasi-interpolation and best approximation. ArXiv e-prints. Preprint arXiv:1505.06931 (2015)
  14. 14.
    Gallistl, D., Huber, P., Peterseim, D.: On the stability of the Rayleigh–Ritz method for eigenvalues. INS Preprint No. 1527. http://peterseim.ins.uni-bonn.de/research/pub/INS1527 (2015)
  15. 15.
    Gaspoz, F.D., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009). doi: 10.1093/imanum/drn039 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Henning, P., Morgenstern, P., Peterseim, D.: Multiscale partition of unity. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations VII. Lecture Notes in Computational Science and Engineering, vol. 100, pp. 185–204. Springer, NewYork (2015)Google Scholar
  17. 17.
    Henning, P., Målqvist, A., Peterseim, D.: Two-level discretization techniques for ground state computations of Bose–Einstein condensates. SIAM J. Numer. Anal. 52(4), 1525–1550 (2014). doi: 10.1137/130921520 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11(4), 1149–1175 (2013). doi: 10.1137/120900332 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hochbruck, M., Sturm, A.: Error analysis of a second order locally implicit method for linear Maxwell’s equations. CRC 1173-Preprint, no. 2015/1, Karlsruher Institut für Technologie. http://www.waves.kit.edu/downloads/CRC1173_Preprint_2015-1 (2015)
  20. 20.
    Joly, P.: Variational methods for time-dependent wave propagation problems. In: Topics in Computational Wave Propagation. Lecture Notes Computation Science Engineering, vol. 31, pp. 201–264. Springer, Berlin (2003)Google Scholar
  21. 21.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Målqvist, A., Peterseim, D.: Computation of eigenvalues by numerical upscaling. Numer. Math. 130(2), 337–361 (2014). doi: 10.1007/s00211-014-0665-6 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comput. 83(290), 2583–2603 (2014). doi: 10.1090/S0025-5718-2014-02868-8 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Målqvist, A., Peterseim, D.: Generalized finite element methods for quadratic eigenvalue problems. ESAIM Math. Model. Numer. Anal. (2016). doi: 10.1051/m2an/2016019 Google Scholar
  25. 25.
    Müller, F.L., Schwab, C.: Finite elements with mesh refinement for wave equations in polygons. J. Comput. Appl. Math. 283, 163–181 (2015). doi: 10.1016/j.cam.2015.01.002 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Oswald, P.: On a BPX-preconditioner for P1 elements. Computing 51(2), 125–133 (1993). doi: 10.1007/BF02243847 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Peterseim, D.: Variational multiscale stabilization and the exponential decay of fine-scale correctors. Preprint arXiv:1505.07611 (2015)
  28. 28.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). doi: 10.2307/2008497 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics. Wiley, Hoboken (1996)zbMATHGoogle Scholar
  30. 30.
    Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987). doi: 10.1093/imanum/7.4.449, http://imajna.oxfordjournals.org/content/7/4/449.abstract

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für Numerische SimulationUniversität BonnBonnGermany

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