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Galerkin Boundary Element Methods for Electromagnetic Scattering

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Topics in Computational Wave Propagation

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 31))

Summary

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.

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References

  1. R. Adams, Sohalev Spaces, Academic Press, New York, 1975.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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. 19–33

    Google Scholar 

  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. 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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Babuŝka, Error bounds for the finite element method, Numer. Math., 16 (1971), pp. 322–333.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, 1991.

    Google Scholar 

  14. A, Buffa, Hodge decompositions on the boundary of a polyhedron: The multiconnected case. Math. Mod. Meth. Appl. Sci., 11 (2001), pp. 1491–1504.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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. 845–876

    Article  MathSciNet  MATH  Google Scholar 

  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. C. Carstensen, A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing, 57 (1996), pp. 301–322.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 1996.

    MATH  Google Scholar 

  25. G. Chen and J. Zhou, Boundary Element Methods, Academic Press, New York, 1992.

    MATH  Google Scholar 

  26. S. Christiansen, Discrete Fredholm properties and convergence estimates for the EFIE, Technical Report 453, CMAP, Ecole Polytechique, Paris, France, 2000.

    Google Scholar 

  27. S. Christiansen, Mixed boundary element method for eddy current problems. Research Report 2002-16, SAM, ETH Zürich, Zürich, Switzeriand, 2002.

    Google Scholar 

  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.

    Article  Google Scholar 

  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. 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. 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. 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. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), pp. 613–626.

    Article  MathSciNet  MATH  Google Scholar 

  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. M. Costabel and M. Dauge, Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci., 22 (1999), pp. 243–258.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Dautray and J.-L, Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Springer, Berlin, 1990.

    Book  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  38. L. Demkowicz., Asymptotic convergence in finite and boundary element methods: Part 1, Theoretical results, Comput. Math. Appl., 27 (1994), pp. 69–84.

    Article  MathSciNet  MATH  Google Scholar 

  39. V. Girauh and P. Raviart, Finite Element Methods For Navier-Stokes Equations, Springer, Berlin, 1986.

    Google Scholar 

  40. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

    MATH  Google Scholar 

  41. P. Grisvard, Singularities in Boundary Value Problems, vol. 22 of Research Notes in Applied Mathematics, Springer, New York, 1992.

    MATH  Google Scholar 

  42. W. Hackbusch, Integral Equations. Theory and Numerical Treatment, vol. 120 of International Series of Numerical Mathematics, Birkhäuser, Basel, 1995.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. R. Hiptmair, Finite elements in computational electromagnetism. Acta Numerica, (2002), pp. 237–339.

    Google Scholar 

  46. R. Hiptmair, Symmetric coupling for eddy current problems, SIAM J. Numer. Anal., 40 (2002), pp. 41–65.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  48. R. Kress, On the boundary operator in electromagnetic scattering, Proc. Royal Soc, Edinburgh, 103A (1986), pp. 91–98

    Article  MathSciNet  Google Scholar 

  49. R, Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, Springer, Berlin, 1989

    Book  MATH  Google Scholar 

  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. R. McCamy and E. Stephan, Solution procedures for three-dimensional eddy-current problems, J. Math. Anal. Appl., 101 (1984), pp. 348–379.

    Article  MathSciNet  Google Scholar 

  52. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.

    MATH  Google Scholar 

  53. J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, vol. 44 of Applied Mathematical Sciences, Springer, Berlin, 2001

    MATH  Google Scholar 

  54. A. Nethe, R. Quast, and H. Stahlmann, Boundary conditions for high frequency eddy current problems, IEEE Trans. Mag., 34(1998), pp. 3331–3334.

    Article  Google Scholar 

  55. L. Paquet, Problemes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse, V. Ser., 4(1982),pp. 103–141.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  58. A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp., 28 (1974), pp. 959–962.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  60. J. Stratton and L. Chu Diffraction, Diffraction theory of electromagnetic waves, Phys. Rev., 56(1939), pp. 99–107.

    Article  Google Scholar 

  61. T. von Petersdorff, Boundary integral equations for mixed Dirichlei. Neumann and transmission problems. Math. Melh. Appl. Sci., 11 (1989), pp. 185–213.

    Article  MATH  Google Scholar 

  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. 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 

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Author information

Authors and Affiliations

  1. Istituto di Matematica applicate e tecnologie informatiche del CNR, Pavia, Italy

    Annalisa Buffa

  2. Seminar für Angewandte Mathematik, ETH Zürich, CH-8092, Zürich, Switzerland

    Ralf Hiptmair

Authors
  1. Annalisa Buffa
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  2. Ralf Hiptmair
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Strathclyde, Richmond Street 26, Glasgow, G1 1XH, UK

    Mark Ainsworth  & Penny Davies  & 

  2. Department of Mathematics, Heriot-Watt University, Riccarton Campus, EH14 4AS, Edinburgh, UK

    Dugald Duncan  & Bryan Rynne  & 

  3. Department of Mathematics and Computer Sciences, Colorado School of Mines, 80401-1887, Golden, USA

    Paul Martin

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Buffa, A., Hiptmair, R. (2003). Galerkin Boundary Element Methods for Electromagnetic Scattering. In: Ainsworth, M., Davies, P., Duncan, D., Rynne, B., Martin, P. (eds) Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55483-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-55483-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00744-9

  • Online ISBN: 978-3-642-55483-4

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