Skip to main content
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

A Correction to this article was published on 29 March 2021

This article has been updated

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Change history

Notes

  1. Quoted from Buffa and Hiptmair, [9], Section 2.2.

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2011)

    MATH  Google Scholar 

  15. Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. 74(5), 1765–1766 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grote, M.J., Keller, J.B.: Nonreflecting boundary conditions for time-dependent scattering. J. Comput. Phys. 127(1), 52–65 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer. 12, 399–450 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Herglotz, G.: Über Potenzreihen mit positivem, reellen Teil im Einheitskreis. Leipz. Ber. 63, 501–511 (1911)

    MATH  Google Scholar 

  21. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  22. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kirsch, A., Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations, Applied Mathematical Sciences, vol. 190. Springer, Berlin (2015)

    MATH  Google Scholar 

  25. Lubich, C.: Convolution quadrature and discretized operational calculus. I. and II. Numer. Math. 52(2), 129–145, 413–425 (1988)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lubich, C.: Convolution quadrature revisited. BIT 44(3), 503–514 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  29. Nédélec, J.-C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35(3), 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Balázs Kovács.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kovács, B., Lubich, C. Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations. Numer. Math. 137, 91–117 (2017). https://doi.org/10.1007/s00211-017-0868-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0868-8

Mathematics Subject Classification

Navigation