Galerkin Boundary Element Methods for Electromagnetic Scattering

  • Annalisa Buffa
  • Ralf Hiptmair
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 31)


Methods based on boundary integral equations are widely used in die numerical simulation of electromagnetic scattering in the frequency domain. This article examines a particular class of these methods, namely the Galerkin boundary element approach, from a theoretical point of view. Emphasis is put on the fundamental differences between acoustic and electromagnetic scattering. The derivation of various boundary integral equations is presented, properties of their discretised counterparts are discussed, and a-priori convergence estimates for the boundary element solutions are rigorously established.

Key words

Electromagnetic scattering boundary integral equations boundary element methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Adams, Sohalev Spaces, Academic Press, New York, 1975.Google Scholar
  2. 2.
    A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H (rot; Ω) and the construction of an extension operator, Manuscripta mathematica, 89 (1996), pp. 159–178.MathSciNetzbMATHGoogle Scholar
  3. 3.
    H. Amman and S. He, Effective impedance boundary conditions for an inhomogeneous thin layer on a curved metallic surface, IEEE Trans. Antennas and Propagation, 46 (1998), pp. 710–715.CrossRefGoogle Scholar
  4. 4.
    H. Ammari and J. Nédélec, Couplage éléments finis-équations intégrales pour la résolution des équations de Maxwell en milieu hétérogene, in Equations aux dérivées partielles et applications. Articles dédies a Jacques-Louis Lions, Gauthier-Villars, Paris, 1998, pp. 19–33Google Scholar
  5. 5.
    H. Ammari and J.-C. Nédélec, Coupling of finite and boundary element methods for the time-harmonic Maxwell equations. II: A symmetric formulation, in The Maz'ya anniversary collection. Vol. 2, J. Rossmann, ed., vol, 110 of Oper, Theory, Adv. Appl., Birkhäuser, Basel, 1999, pp. 23–32.Google Scholar
  6. 6.
    H. Ammari and J.-C. Nédélec. Coupling integral equations method and finite volume elements for the resolution of the Leontovich boundary value problem for the time-harmonic Maxwell equations in three dimensional heterogeneous media, in Mathematical aspects of boundary element methods. Minisymposium during the lABEM 98 conference, dedicated to Vladimir Maz'ya on the occasion of his 60th birthday on 31st December 1997, M. Bonnet, ed., vol. 4 i of CRC Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2000, pp. 11–22.Google Scholar
  7. 7.
    C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Appl. Sci., 21 (1998), pp. 823–864.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I. Babuŝka, Error bounds for the finite element method, Numer. Math., 16 (1971), pp. 322–333.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Bendali, Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Partial differential equations: Theory and numerical solution. Proceedings of the ICM'98 satellite conference, Prague. Czech Republic, August 10–16, 1998., W. Jäger, ed., vol. 406 of CRC Res. Notes Math., Boca Raton, FL, 2000, Chapman & Hall/CRC, pp. 10–24.Google Scholar
  10. 10.
    A. Bendali and L. Vernhet, The Leontovich boundary value problem and its boundary integral equations solution. Preprint, CNRS-UPS-INSA, Department de Genie Mathematique, Toulouse, France, 2001.Google Scholar
  11. 11.
    H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.zbMATHGoogle Scholar
  12. 12.
    F. Brezzi, J. Douglas, and D. Marini, Two families of mixed finite elements for 2nd order elliptic problems, Numer. Math., 47 (1985), pp. 217–235.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, 1991.Google Scholar
  14. 14.
    A, Buffa, Hodge decompositions on the boundary of a polyhedron: The multiconnected case. Math. Mod. Meth. Appl. Sci., 11 (2001), pp. 1491–1504.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Buffa, Traces theorems for functional spaces related to Maxwell equations: An overwiew. To appear in Proceedings of the GAMM Workshop on Computational Electromagnetics, Kiel, January 26th — 28th, 2001.Google Scholar
  16. 16.
    A. Buffa and S. Christiansen, The electric field integral equation on Lipschitz screens: Definition and numerical approximation, Numer. Mathem. (2002), DOI 10.1007/s00211-002-0422-0.Google Scholar
  17. 17.
    A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell’s equations. Part I: An integration by parts formula in Lipschitz polyhedra., Math. Meth. Appl. Sci., 24 (2001), pp. 9–30.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell’s equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Medi. Appl. Sci., 21 (2001), pp. 31–48.MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Buffa, M. Costabel, and C. Schwab, Boundary element methods for Maxwell’s equations on non-smooth domains, Numer. Mathem. 92 (2002) 4, pp. 679–710.CrossRefGoogle Scholar
  20. 20.
    A. Buffa, M. Costabel, and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl., 276/2 (2002), pp. 845–876MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Buffa, R. Hiptmair, T von Petersdorif, and C. Schwab, Boundary element methods for Maxwell equations on Lipschitz domains, Numer. Math., (2002). To appear.Google Scholar
  22. 22.
    C. Carstensen, A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing, 57 (1996), pp. 301–322.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    C. Carstensen and P. Wriggers, On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements, IMA J. Numer. Anal., 17 (1997), pp. 201–238.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 1996.zbMATHGoogle Scholar
  25. 25.
    G. Chen and J. Zhou, Boundary Element Methods, Academic Press, New York, 1992.zbMATHGoogle Scholar
  26. 26.
    S. Christiansen, Discrete Fredholm properties and convergence estimates for the EFIE, Technical Report 453, CMAP, Ecole Polytechique, Paris, France, 2000.Google Scholar
  27. 27.
    S. Christiansen, Mixed boundary element method for eddy current problems. Research Report 2002-16, SAM, ETH Zürich, Zürich, Switzeriand, 2002.Google Scholar
  28. 28.
    S. Christiansen and J.-C. Nédélec, Des préondilionneurs pour la résolution numérique des équations intégrales de frontière de l'electromagnétisme, CR. Acad. Sci. Paris, Ser. I Math, 31 (2000), pp. 617–622.CrossRefGoogle Scholar
  29. 29.
    S. Christiansen and J.-C. Nédélec, A preconditioner for the electric field integral equation based on Calderón formulae. To appear in STAM J. Numer. Anal.Google Scholar
  30. 30.
    P. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4 of Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1978.Google Scholar
  31. 31.
    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer, Heidelberg, 2nd ed., 1998.Google Scholar
  32. 32.
    M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements, in Boundary Elements TX, C. Brebbia, W. Wendland, and G. Kuhn, eds.. Springer, Berlin, 1987, pp. 411–420.Google Scholar
  33. 33.
    M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), pp. 613–626.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains, in Analysis, Numerics and Apphcations of Differential and Integral Equations, M. Bach, ed., vol. 379 of Longman Pitman Res. Notes Math. Ser., Addison Wesley, Hariow, 1998, pp. 69–76.Google Scholar
  35. 35.
    M. Costabel and M. Dauge, Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci., 22 (1999), pp. 243–258.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    R. Dautray and J.-L, Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Springer, Berlin, 1990.CrossRefGoogle Scholar
  37. 37.
    A. de La Bourdonnaye, Some formulations coupling finite element amd integral equation method for Helmholtz equation and electromagnetism, Numer. Math., 69 (1995), pp. 257–268.MathSciNetCrossRefGoogle Scholar
  38. 38.
    L. Demkowicz., Asymptotic convergence in finite and boundary element methods: Part 1, Theoretical results, Comput. Math. Appl., 27 (1994), pp. 69–84.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    V. Girauh and P. Raviart, Finite Element Methods For Navier-Stokes Equations, Springer, Berlin, 1986.Google Scholar
  40. 40.
    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.zbMATHGoogle Scholar
  41. 41.
    P. Grisvard, Singularities in Boundary Value Problems, vol. 22 of Research Notes in Applied Mathematics, Springer, New York, 1992.zbMATHGoogle Scholar
  42. 42.
    W. Hackbusch, Integral Equations. Theory and Numerical Treatment, vol. 120 of International Series of Numerical Mathematics, Birkhäuser, Basel, 1995.zbMATHGoogle Scholar
  43. 43.
    C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell’s equations, SIAM J. Math. Anal., 27 (1996), pp. 1597–1630.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    R. Hiptmair, Coupling of finite elements and boundary elements in electromagnetic scattering, Report 164, SFB 382, Universität Tübingen, Tübingen, Germany, July 2001. Submitted to SIAM J. Numer. Anal.Google Scholar
  45. 45.
    R. Hiptmair, Finite elements in computational electromagnetism. Acta Numerica, (2002), pp. 237–339.Google Scholar
  46. 46.
    R. Hiptmair, Symmetric coupling for eddy current problems, SIAM J. Numer. Anal., 40 (2002), pp. 41–65.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    R. Hiptmair and C. Schwab, Natural boundary element methods for the electric field integral equation on polyhedra, SIAM J. Numer. Anal., 40 (2002), pp. 66–86.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    R. Kress, On the boundary operator in electromagnetic scattering, Proc. Royal Soc, Edinburgh, 103A (1986), pp. 91–98MathSciNetCrossRefGoogle Scholar
  49. 49.
    R, Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, Springer, Berlin, 1989CrossRefzbMATHGoogle Scholar
  50. 50.
    M. Kuhn and O. Steinbach, FEM-BEM coupling for 3d exterior magnetic field problems, Math. Meth. Appl. Sci., (2002). To appear.Google Scholar
  51. 51.
    R. McCamy and E. Stephan, Solution procedures for three-dimensional eddy-current problems, J. Math. Anal. Appl., 101 (1984), pp. 348–379.MathSciNetCrossRefGoogle Scholar
  52. 52.
    W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.zbMATHGoogle Scholar
  53. 53.
    J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, vol. 44 of Applied Mathematical Sciences, Springer, Berlin, 2001zbMATHGoogle Scholar
  54. 54.
    A. Nethe, R. Quast, and H. Stahlmann, Boundary conditions for high frequency eddy current problems, IEEE Trans. Mag., 34(1998), pp. 3331–3334.CrossRefGoogle Scholar
  55. 55.
    L. Paquet, Problemes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse, V. Ser., 4(1982),pp. 103–141.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    P. A. Raviari and J. M. Thomas, A Mixed Finite Element Method for Second Order Elliptic Problems, vol. 606 of Springer Lecture Notes in Mathematics, Springer, Ney York, 1977, pp. 292–315.Google Scholar
  57. 57.
    M. Reissel, On a transmission boundary-value problem for the time-harmonic Maxwell equations without displacement currents, SLAM J. Math. Anal., 24 (1993), pp. 1440–1457.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp., 28 (1974), pp. 959–962.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    O. Steinbach and W. Wendland, The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math., 9 (1998), pp. 191–216.MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    J. Stratton and L. Chu Diffraction, Diffraction theory of electromagnetic waves, Phys. Rev., 56(1939), pp. 99–107.CrossRefGoogle Scholar
  61. 61.
    T. von Petersdorff, Boundary integral equations for mixed Dirichlei. Neumann and transmission problems. Math. Melh. Appl. Sci., 11 (1989), pp. 185–213.CrossRefzbMATHGoogle Scholar
  62. 62.
    W. Wendland, Boundary element methods for elliptic problems, in Mathematical Theory of Finite and Boundary Element Methods, A. Schatz, V. Thomée, and W. Wendland, eds., vol. 15 of DMV-Seminar, Birkhäuser, Basel, 1990, pp. 219–276.Google Scholar
  63. 63.
    J. Xu and L. Zikatanov, Some observations on Babuska and Brezzi theories. Report AM222, PennState Department of Mathematics, College Park, PA, September 2000. To appear in Numer. Math.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Annalisa Buffa
    • 1
  • Ralf Hiptmair
    • 2
  1. 1.Istituto di Matematica applicate e tecnologie informatiche del CNRPaviaItaly
  2. 2.Seminar für Angewandte MathematikETH ZürichZürichSwitzerland

Personalised recommendations