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Numerische Mathematik

, Volume 132, Issue 3, pp 569–595 | Cite as

Adaptive time discretization for retarded potentials

  • S. SauterEmail author
  • A. Veit
Article

Abstract

In this paper, we will present advanced discretization methods for solving retarded potential integral equations. We employ a \(C^{\infty }\)-partition of unity method in time and a conventional boundary element method for the spatial discretization. One essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix. We present here an approach which is based on quadrature for (non-analytic) \(C^{\infty }\) functions in combination with certain Chebyshev expansions. Furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement. Numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process.

Mathematics Subject Classification

35L05 65N38 65R20 

Notes

Acknowledgments

The second author gratefully acknowledges the support given by the Swiss National Science Foundation (No. P2ZHP2_148705).

References

  1. 1.
    Bamberger, A., Duong, T.H.: Formulation Variationnelle Espace-Temps pur le Calcul par Potientiel Retardé de la Diffraction d’une Onde Acoustique. Math. Methods Appl. Sci. 8, 405–435 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47, 227–249 (2008)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Ding, Y., Forestier, A., Duong, T.H.: A Galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface. J. Acoust. Soc. Am. 86(4), 1566–1572 (1989)CrossRefGoogle Scholar
  4. 4.
    Faermann, B.: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods: part I. The two-dimensional case. IMA J. Numer. Anal. 20, 203–234 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Faermann, B.: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. Part II. The three-dimensional case. Numer. Math. 92(3), 467–499 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Friedman, M., Shaw, R.: Diffraction of pulses by cylindrical obstacles of arbitrary cross section. J. Appl. Mech. 29, 40–46 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Gläfke, M.: Adaptive methods for time domain boundary integral equations. PhD thesis, Brunel University (2013)Google Scholar
  8. 8.
    Ha-Duong, T.: On retarded potential boundary integral equations and their discretisation. In: Topics in computational wave propagation: direct and inverse problems, vol. 31 of Lect. Notes Comput. Sci. Eng, pp. 301–336. Springer, Berlin (2003)Google Scholar
  9. 9.
    Ha-Duong, T., Ludwig, B., Terrasse, I.: A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Int. J. Numer. Methods Eng. 57, 1845–1882 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Khoromskij, B., Sauter, S., Veit, A.: Fast quadrature techniques for retarded potentials based on TT/QTT tensor approximation. Comput. Methods Appl. Math. 11(3), 342–362 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    López-Fernández, M., Sauter, S.A.: Generalized convolution quadrature with variable time stepping. Part II: algorithms and numerical results. Technical Report 09–2012, Institut für Mathematik, Universität Zürich (2012)Google Scholar
  12. 12.
    López-Fernández, M., Sauter, S.: Generalized convolution quadrature with variable time stepping. IMA J. Numer. Anal. 33(4), 1153–1175 (2013)Google Scholar
  13. 13.
    Lubich, C.: Convolution quadrature and discretized operational calculus I. Numerische Mathematik 52, 129–145 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Nédélec, J., Abboud, T., Volakis, J.: Stable solution of the retarded potential equations, Applied Computational Electromagnetics Society (ACES) Symposium Digest. In: 17th Annual Review of Progress, Monterey (2001)Google Scholar
  15. 15.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in C: the art of scientific computing. 2nd edn. Cambridge University Press, New York (1992)Google Scholar
  16. 16.
    Sauter, S., Schwab, C.: Boundary element methods. Springer Series in Computational Mathematics. Springer, Berlin (2010)Google Scholar
  17. 17.
    Sauter, S., Veit, A.: Retarded boundary integral equations on the sphere: exact and numerical solution. IMA J. Numer. Anal. (2013). doi: 10.1093/imanum/drs059
  18. 18.
    Sauter, S., Veit, A.: A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions. Numerische Mathematik 123(1), 145–176 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Schmid, M.: Analysis of tenor gaussian quadrature of functions of class \({C}^\infty \). Master’s thesis, University of Zurich (2013)Google Scholar
  20. 20.
    Trefethen, L.: Is gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Weile, D.S., Pisharody, G., Chen, N.W., Shanker, B., Michielssen, E.: A novel scheme for the solution of the time-domain integral equations of electromagnetics. IEEE Trans. Antennas Propag. 52, 283–295 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Department of Computer ScienceUniversity of ChicagoChicagoUSA

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