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Numerische Mathematik

, Volume 140, Issue 2, pp 427–447 | Cite as

Super-convergence and post-processing for mixed finite element approximations of the wave equation

  • Herbert Egger
  • Bogdan Radu
Article
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Abstract

We consider the numerical approximation of acoustic wave propagation problems by mixed \(\text {BDM}_{k+1}\)–\(\text {P}_k\) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank–Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore also apply for non-convex domains and non-uniform meshes.

Mathematics Subject Classification

35L05 35L50 65L20 65M60 
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Notes

Acknowledgements

The authors are grateful for financial support by the German Research Foundation (DFG) via Grants IRTG 1529 and TRR 154 project C4, and by the “Excellence Initiative” of the German Federal and State Governments via the Graduate School of Computational Engineering GSC 233 at Technische Universität Darmstadt.

References

  1. 1.
    Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19, 7–32 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baker, G.A.: Error estimates for finite element approximations for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 564–576 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, G.A., Bramble, J.H.: Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO Anal. Numer. 13, 75–100 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bangerth, W., Rannacher, R.: Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoust. 1, 1–17 (1999)zbMATHGoogle Scholar
  5. 5.
    Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, vol. 1939. Springer, Fondazione C.I.M.E., Berlin, Florence (2008)CrossRefGoogle Scholar
  6. 6.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Brezzi, F., Douglas, J., Duràn, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 47, 237–250 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 88, 217–235 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Y.: Global superconvergence for a mixed finite element method for the wave equation. Syst. Sci. Math. Sci. 12, 159–165 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Huan, Y.: The superconvergence of mixed finite element methods for nonlinear hyperbolic equations. Numer. Simul. 3, 155–158 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Z.: Superconvergence results for Galerkin methods for wave propagation in various porous media. Numer. Methods Partial Differ. Equ. 12, 99–122 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cohen, G.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Cohen, G., Joly, O., Roberts, J.E., Tordjman, N.: Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38, 2047–2078 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions. SIAM J. Numer. Anal. 33, 492–504 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Douglas, J., Dupont, T., Wheeler, M.F.: A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations. Math. Comput. 32, 345–362 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dupont, T.: \(l^2\) estimates for Galerkin methods for second-order hyperbolic equations. SIAM J. Numer. Anal. 10, 880–889 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  18. 18.
    Geveci, T.: On the application of mixed finite element methods to the wave equations. RAIRO Model. Mathods Anal. Numer. 22, 243–250 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jenkins, E.W., Rivière, T., Wheeler, M.F.: A priori error estimates for mixed finite element approximations of the acoustic wave equation. SIAM J. Numer. Anal. 40, 1698–1715 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Joly, P.: Variational methods for time-dependent wave propagation problems. In: Ainsworth, M., Davies, P., Duncan, D., Martin, P., Rynne, B. (eds.) Topics in Computational Wave Propagation, LNCSE, vol. 31, pp. 201–264. Springer, BerlinGoogle Scholar
  21. 21.
    Karaa, S.: Error estimates for finite element approximations of a viscous wave equation. Numer. Func. Anal. Optim. 32, 750–767 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Landau, L.D., Lifshitz, E.M.: Course of theoretical physics, vol. 6, 2nd edn. Pergamon Press, Oxford, 1987. Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. ReidGoogle Scholar
  23. 23.
    Makridakis, C.G.: On mixed finite element methods for linear elastodynamics. Numer. Math. 61, 235–260 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Makridakis, C.G., Monk, P.: Time-discrete finite element schemes for Maxwell’s equations. RAIRO Model. Mathods Anal. Numer. 29, 171–197 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Monk, P.: Analysis of a finite element methods for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  27. 27.
    Schöberl, J.: Commuting quasi-interpolation operators for mixed finite elements. Isc-01-10-math, Institute for Scientific Computing, Texas A&M University (2001)Google Scholar
  28. 28.
    Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Model. Mathods Anal. Numer. 25, 151–167 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, F., Chen, Y., Tang, Y.: Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations. Numer. Methods Partial Differ. Equ. 30, 175–186 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, X., Bathe, K.-J.: Displacement/pressure based mixed finite element formulations for acoustic fluid–structure interaction problems. Int. J. Numer. Methods Eng. 40, 2001–2017 (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTU DarmstadtDarmstadtGermany
  2. 2.Graduate School for Computational EngineeringTU DarmstadtDarmstadtGermany

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