Numerische Mathematik

, Volume 137, Issue 1, pp 91–117 | Cite as

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

  • Balázs KovácsEmail author
  • Christian Lubich


Maxwell’s equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a weak first-order formulation of the continuous equations. The method proposed here uses a discontinuous Galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary. The method is explicit in the interior and implicit on the boundary. Stability and convergence of the spatial semidiscretization are proven, and with a computationally simple stabilization term, this is also shown for the full discretization.

Mathematics Subject Classification

35Q61 65M60 65M38 65M12 65R20 



We thank two anonymous referees for their helpful comments. We are grateful for the helpful discussions on spatial discretizations with Ralf Hiptmair (ETH Zürich) during a BIRS Workshop (16w5071) in Banff. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB 1173.


  1. 1.
    Abboud, T., Joly, P., Rodríguez, J., Terrasse, I.: Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains. J. Comput. Phys. 230(15), 5877–5907 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballani, J., Banjai, L., Sauter, S., Veit, A.: Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature. Numer. Math. 123(4), 643–670 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments. SIAM J. Sci. Comput. 32(5), 2964–2994 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banjai, L., Lubich, C., Melenk, J.M.: Runge–Kutta convolution quadrature for operators arising in wave propagation. Numer. Math. 119(1), 1–20 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banjai, L., Lubich, C., Sayas, F.-J.: Stable numerical coupling of exterior and interior problems for the wave equation. Numer. Math. 129, 611–646 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brenner, S., Li, F., Sung, L.-Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comput. 76(258), 573–595 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buffa, A., Costabel, M., Sheen, D.: On traces for \(H(\text{ curl }, \varOmega )\) in Lipschitz domains. J. Math. Anal. Appl. 276(2), 845–867 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buffa, A., Hiptmair, R.: Galerkin boundary element methods for electromagnetic scattering. In: Ainsworth, M., Davies, P., Duncan, D., Rynne, B., Martin, P. (eds.) Topics in Computational Wave Propagation, pp. 83–124. Springer, Berlin (2003)Google Scholar
  10. 10.
    Buffa, A., Hiptmair, R., von Petersdorff, T., Schwab, C.: Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95(3), 459–485 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, Q., Monk, P.: Introduction to applications of numerical analysis in time domain computational electromagnetism. In: Blowey, J., Jensen, M. (eds.) Frontiers in Numerical Analysis, pp. 149–225. Springer, Berlin (2012)Google Scholar
  12. 12.
    Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194(2), 588–610 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Costabel, M.: Time-dependent problems with the boundary integral equation method. In: Encyclopedia of Computational Mechanics, Wiley (2004). doi: 10.1002/0470091355.ecm022
  14. 14.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2011)zbMATHGoogle Scholar
  15. 15.
    Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. 74(5), 1765–1766 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grote, M.J., Keller, J.B.: Nonreflecting boundary conditions for time-dependent scattering. J. Comput. Phys. 127(1), 52–65 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hagstrom, T., Mar-Or, A., Givoli, D.: High-order local absorbing conditions for the wave equation: extensions and improvements. J. Comput. Phys. 227(6), 3322–3357 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer. 12, 399–450 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Herglotz, G.: Über Potenzreihen mit positivem, reellen Teil im Einheitskreis. Leipz. Ber. 63, 501–511 (1911)zbMATHGoogle Scholar
  21. 21.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  22. 22.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hochbruck, H., Sturm, A.: Error analysis of a second order locally implicit method for linear Maxwell’s equations. SIAM J. Numer. Anal. 54(5), 3167–3191 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kirsch, A., Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations, Applied Mathematical Sciences, vol. 190. Springer, Berlin (2015)zbMATHGoogle Scholar
  25. 25.
    Lubich, C.: Convolution quadrature and discretized operational calculus. I. and II. Numer. Math. 52(2), 129–145, 413–425 (1988)Google Scholar
  26. 26.
    Lubich, C.: On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numer. Math. 67(3), 365–389 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lubich, C.: Convolution quadrature revisited. BIT 44(3), 503–514 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRefzbMATHGoogle Scholar
  29. 29.
    Nédélec, J.-C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35(3), 315–341 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nédélec, J.C.: Approximation of integral equations by finite elements. Error analysis. In: Dautray, R., Lions, J.-L. (eds.) Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Chapter XIII, pp. 359–370. Springer, Berlin (1990)Google Scholar
  31. 31.
    Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani I., Magenes E. (eds.) Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer, Berlin (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversity of TübingenTübingenGermany

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