Skip to main content
SpringerLink
Log in

Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using Runge–Kutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimates.

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

Similar content being viewed by others

References

  1. Abboud, T., Nédélec, J., Volakis, J.: Stable solution of the retarded potential equations. In: Proceedings of 17th Annual Review Progress in Applied Computer and Electromagnetics, Monterey, CA, March 2001, pp. 146–151 (2001)

  2. Bachelot, A., Bounhoure, L., Pujols, A.: Couplage éléments finis-potentiels retardés pour la diffraction électromagnétique par un obstacle hétérogène. Numer. Math. 89, 257–306 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bamberger, A., Duong, T.H.: Formulation Variationnelle Espace-Temps pur le Calcul par Ptientiel Retardé de la Diffraction d’une Onde Acoustique. Math. Meth. Appl. Sci. 8, 405–435 (1986)

    Article  MATH  Google Scholar 

  4. Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: Algorithms and experiments. SIAM J. Sci. Comput. 32(5), 2964–2994 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banjai, L., Lubich, C.: An error analysis of Runge-Kutta convolution quadrature. BIT 51(3), 483–496 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banjai, L., Lubich, C., Melenk, J.M.: A refined convergence analysis of Runge-Kutta convolution quadrature. Numer. Math. 119(1), 1–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Banjai, L., Sauter, S.: Frequency Explicit Regularity Estimates for the Electric Field Integral Operator. Preprint 05–2012, Universität Zürich (2012)

  8. Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47, 227–249 (2008)

    Article  MathSciNet  Google Scholar 

  9. Banjai, L., Schanz, M.: Wave propagation problems treated with convolution quadrature and bem. In: Langer, U., Schanz, M., Steinbach, O., Wendland, W.L. (eds.) Fast Boundary Element Methods in Engineering and Industrial Applications, Lecture Notes in Applied and Computational Mechanics, vol. 63, pp. 145–184. Springer, Berlin (2012)

  10. Bendali, A.: Numerical analysis of the exterior boundary value problem for the time-harmonic maxwell equations by a boundary finite element method part 2: the discrete problem. Math. Comput. 43, 47–68 (1984)

    MathSciNet  MATH  Google Scholar 

  11. Buffa, A., Ciarlet, P.: On traces for functional spaces related to Maxwell’s equations. part I: an integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24, 9–30 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buffa, A., Ciarlet, P. Jr.: On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci. 24(1), 31–48 (2001)

    Google Scholar 

  13. Buffa, A., Costabel, M., Sheen, D.: On traces for H(curl,\({\Omega }\)) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–867 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, Q., Monk, P., Wang, X., Weile, D.: Analysis of convolution quadrature applied to the time-domain electric field integral equation. Commun. Comput. Phys. 11, 383–399 (2012)

    Google Scholar 

  15. Chew, W.C., Jin, J.-M., Michielssen, E., Song, J.M.: Fast and Efficient Algorithms in Computational Electromagnetics. Artech House, Boston, London (2001)

    Google Scholar 

  16. Christiansen, S.H.: Uniformly stable preconditioned mixed boundary element method for low-frequency electromagnetic scattering. C.R. Math. Acad. Sci. Paris 336(8), 677–680 (2003)

  17. Erma, V.A.: Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case. Phys. Rev. (2) 176, 1544–1553 (1968)

    Google Scholar 

  18. Friedman, M., Shaw, R.: Diffraction of pulses by cylindrical obstacles of arbitrary cross section. J. Appl. Mech. 29, 40–46 (1962)

    Google Scholar 

  19. Ha-Duong, T.: On retarded potential boundary integral equations and their discretisation. In: Topics in Computational Wave Propagation: Direct and Iverse Problems, Lect. Notes Comput. Sci. Eng., vol. 31, pp. 301–336. Springer, Berlin (2003)

  20. Hackbusch, W., Kress, W., Sauter, S.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation by cutoff and panel-clustering. In: Schanz, M., Steinbach, O. (eds.) Boundary Element Analysis, pp. 113–134. Springer, Berlin (2007)

    Chapter  Google Scholar 

  21. Hackbusch, W., Kress, W., Sauter, S.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. IMA. J. Numer. Anal. 29, 158–179 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hiptmair, R., Schwab, C.: Natural Boundary Element Methods for the Electric Field Integral Equation on Polyhedra. SIAM J. Numer. Anal. 40, 66–86 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kress, W., Sauter, S.: Numerical treatment of retarded boundary integral equations by sparse panel clustering. IMA J. Numer. Anal. 28(1), 162–185 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kriemann, R.: HLIBpro C language interface. Technical Report 10/2008, MPI for Mathematics in the Sciences, Leipzig (2008)

  25. Kriemann, R.: HLIBpro user manual. Technical Report 9/2008, MPI for Mathematics in the Sciences, Leipzig (2008)

  26. Lubich, C.: Convolution quadrature and discretized operational calculus I. Numerische Mathematik 52, 129–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lubich, C.: Convolution quadrature and discretized operational calculus II. Numerische Mathematik 52, 413–425 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lubich, C.: On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numerische Mathematik 67(3), 365–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lubich, C.: Convolution quadrature revisited. BIT. Numer. Math. 44, 503–514 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lubich, C., Ostermann, A.: Runge-Kutta methods for parabolic equations and convolution quadrature. Math. Comp. 60(201), 105–131 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nédélec, J.: Acoustic and Electromagnetic Equations. Springer, Berlin (2001)

    MATH  Google Scholar 

  32. Pujols, A.: Equations intégrales Espace-Temps pour le système de Maxwell Application au calcul de la Surface Equivalente Radar. PhD thesis, L’Universite Bordeaux I (1991)

  33. Pujols, A.: Time Dependent Integral Method for Maxwell Equations. Rapport CESTA/SI A. P. 6589 (1991)

  34. Rao, S., Wilton, D., Glisson, A.: Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antennas Propagat. AP 30, 409–418 (1982)

    Article  Google Scholar 

  35. Raviart, P., Thomas, J.: A mixed finite element method for 2-nd order elliptic problems. Math. Aspects Finite Element Methods (Springer Lecture Notes in Mathematics) 606, 292–315 (1977)

    Article  MathSciNet  Google Scholar 

  36. Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Berlin (2010)

    Google Scholar 

  37. Sauter, S., Veit, A.: A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions. Numer. Math. 1–32, (2012). doi:10.1007/s00211-012-0483-7

  38. Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28(2), 421–438 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Schanz, M.: Wave propagation in viscoelastic and poroelastic continua: a boundary element approach. In: Lecture Notes in Applied and Computational Mechanics, vol. 2. Springer, Berlin (2001)

  40. Terrasse, I.: Résolution mathématique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés. PhD thesis, Ecole polytechnique, 1993

  41. Veit, A.: Convolution quadrature for time-dependent Maxwell equations. University of Zurich, Master’s thesis (2009)

    Google Scholar 

  42. Wang, X., Weile, D.: Implicit Runge-Kutta methods for the discretization of time domain integral equations. IEEE Trans. Antennas Propagat. 59(12), 4651–4663 (2011)

    Article  MathSciNet  Google Scholar 

  43. Wang, X., Wildman, R., Weile, D., Monk, P.: A finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics. IEEE Trans. Antennas Propagat. 56(8), 2442–2452 (2008)

    Article  MathSciNet  Google Scholar 

  44. Weile, D., Ergin, A., Shanker, B., Michielssen, E.: An accurate discretization scheme for the numerical solution of time domain integral equations. IEEE Antennas Propagat. Soc. Int. Symp. 2, 741–744 (2000)

    Google Scholar 

  45. Weile, D., Pisharody, G., Chen, N., Shanker, B., Michielssen, E.: A novel scheme for the solution of the time-domain integral equations of electromagnetics. IEEE Trans. Antennas Propagat. 52, 283–295 (2004)

    Article  Google Scholar 

  46. Weile, D., Shanker, B., Michielssen, E.: An accurate scheme for the numerical solution of the time domain electric field integral equation. IEEE Antennas Propagat. Soc. Int. Symp. 4, 516–519 (2001)

    Google Scholar 

  47. Wildman, A., Pisharody, G., Weile, D., Balasubramaniam, S., Michielssen, E.: An accurate scheme for the solution of the time-domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation. IEEE Trans. Antennas Propagat. 52, 2973–2984 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The second author gratefully acknowledges the helpful discussions he had with Qiang Chen while visiting University of Delaware. The fourth author gratefully acknowledges the support given by SNF, No. PDFMP2_127437/1.

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22-26, 04103,  Leipzig, Germany

    J. Ballani & L. Banjai

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057,  Zürich, Switzerland

    S. Sauter & A. Veit

Authors
  1. J. Ballani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Banjai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Sauter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Veit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Sauter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ballani, J., Banjai, L., Sauter, S. et al. Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature. Numer. Math. 123, 643–670 (2013). https://doi.org/10.1007/s00211-012-0503-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-012-0503-7

Mathematics Subject Classification

Search

Navigation