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Differential Forms and Boundary Integral Equations for Maxwell-Type Problems

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Fast Boundary Element Methods in Engineering and Industrial Applications

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 63))

Abstract

We present boundary-integral equations for Maxwell-type problems in a differential-form setting. Maxwell-type problems are governed by the differential equation (δd − k 2)ω = 0, where k ∈ ℂ holds, subject to some restrictions. This problem class generalizes curl curl- and grad-types of problems in three dimensions. The goal of the paper is threefold: 1) Establish the Sobolev-space framework in the full generality of differential-form calculus on a smooth manifold of arbitrary dimension and with Lipschitz boundary. 2) Introduce integral transformations and fundamental solutions, and derive a representation formula for Maxwell-type problems. 3) Leverage the power of differential-form calculus to gain insight into properties and inherent symmetries of boundary-integral equations of Maxwell-type.

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References

  1. Ammari, H., Buffa, A., Nédélec, J.C.: A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60(5), 1805–1823 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bossavit, A.: On the representation of differential forms by potentials in dimension 3. In: van Rienen, U., Günther, M., Hecht, D. (eds.) Scientific Computing in Electrical Engineering. LNCSE, vol. 18, pp. 97–104. Springer, Berlin (2001)

    Chapter  Google Scholar 

  4. Buffa, A.: Hodge decompositions on the boundary of a polyhedron: The non-simply connected case. Math. Models Meth. Appl. Sciences 11(9), 1491–1503 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buffa, A.: Trace theorems on non-smooth boundaries for functional spaces related to Maxwell equations: an overview. In: Carstensen, C., et al. (eds.) Computational Electromagnetics. LNCSE, vol. 28, pp. 23–34. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  6. Buffa, A.: Remarks on the discretization of some nonpositive operators with application to heterogeneous Maxwell problems. SIAM J. Numer. Anal. 43(1), 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buffa, A., Christiansen, S.: A dual finite element complex on the barycentric refinement. Math. Comp. 76(260), 1743–1769 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buffa, A., Ciarlet Jr., P.: 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, 9–30 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Buffa, A., Costabel, M., Schwab, C.: Boundary element methods for Maxwell equations in non-smooth domains. Numer. Math. 92(4), 679–710 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Buffa, A., Hiptmair, R., von Petersdorff, T., Schwab, C.: Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95, 459–485 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Christiansen, S.: Mixed boundary element method for eddy current problems. Tech. Rep. 2002-16, Seminar für Angewandte Mathematik, ETH Zürich (2002)

    Google Scholar 

  14. Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. Springer, Berlin (1998)

    MATH  Google Scholar 

  15. Greub, W.: Multilinear Algebra. Springer, Berlin (1967)

    MATH  Google Scholar 

  16. Hehl, F., Obukhov, Y.: Foundations of Classical Electrodynamics. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40(1), 41–65 (2003)

    Article  MathSciNet  Google Scholar 

  19. Hiptmair, R.: Boundary element methods for eddy current computation. In: Boundary Element Analysis. LNACM, vol. 29, pp. 213–248. Springer, Berlin (2007)

    Chapter  Google Scholar 

  20. Jänich, K.: Vector Analysis. Springer, Berlin (2001)

    Google Scholar 

  21. Jawerth, B., Mitrea, M.: Higher dimensional electromagnetic scattering theory on C 1 and Lipschitz domains. American J. Math. 117(4), 929–963 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kotiuga, R.: Weitzenböck identities and variational formulations in nanophotonics and micromagnetics. IEEE Trans. Magn. 43(4), 1669–1672 (2007)

    Article  Google Scholar 

  23. Kress, R.: Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebiger Dimension und beliebigen Ranges. Arch. Ration. Mech. Anal. 47, 59–80 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kurz, S., Auchmann, B., Flemisch, B.: Dimensional reduction of field problems in a differential-form framework. COMPEL 28(4), 907–921 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Kythe, P.K.: Fundamental Solutions for Differential Operators and Applications. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

  26. Maue, A.W.: Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Z. Physik A 126, 601–618 (1949)

    MathSciNet  MATH  Google Scholar 

  27. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  28. Nédélec, J.C.: Integral equations with non integrable kernels. Integral Eqn. Oper. Th. 5, 562–572 (1982)

    Article  MATH  Google Scholar 

  29. Pauly, D.: Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains. Adv. Math. Sci. Appl. 16(2), 591–622 (2006)

    MathSciNet  MATH  Google Scholar 

  30. de Rham, G.: Differentiable Manifolds. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  31. Rota, G.C.: Indiscrete Thoughts. Birkhäuser, Boston (2000)

    Google Scholar 

  32. Stratton, J.: Electromagnetic Theory. McGraw-Hill Book Company, New York (1941)

    MATH  Google Scholar 

  33. Tai, C.T.: Dyadic Green’s functions in electromagnetic theory. IEEE Press, Piscataway (1994)

    Google Scholar 

  34. Tucker, R.W.: Differential form valued forms and distributional electromagnetic sources. J. Math. Phys. 50(3) (2009)

    Google Scholar 

  35. Warnick, K., Arnold, D.: Electromagnetic Green functions using differential forms. J. Electromagn. Waves Appl. 10(3), 427–438 (1996)

    Article  Google Scholar 

  36. Weck, N.: Traces of differential forms on Lipschitz boundaries. Internat. Math. J. Anal. Appl. 24(2), 147–169 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Weck, N., Witsch, K.: Generalized spherical harmonics and exterior differentiation in weighted Sobolev spaces. Math. Meth. Appl. Sci. 17(13), 1017–1043 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Weyl, H.: Die natürlichen Randwertaufgaben im Außenraum für Strahlungsfelder beliebiger Dimension und beliebigen Ranges. Math. Z. 56(2), 105–119 (1952)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electronics, Tampere University of Technology, 33101, Tampere, Finland

    Stefan Kurz

  2. CERN/TE, Geneva, 1211, Switzerland

    Bernhard Auchmann

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  1. Stefan Kurz
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  2. Bernhard Auchmann
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Correspondence to Stefan Kurz .

Editor information

Editors and Affiliations

  1. Inst. Numerische Mathematik, Universität Linz, Altenbergerstr. 69, Linz, 4040, Austria

    Ulrich Langer

  2. Inst. Allgemeine Mechanik, TU Graz, Technikerstr. 4, Graz, 8010, Austria

    Martin Schanz

  3. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, Graz, 8010, Austria

    Olaf Steinbach

  4. Institut für Angewandte Analysis, und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Wolfgang L. Wendland

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Kurz, S., Auchmann, B. (2012). Differential Forms and Boundary Integral Equations for Maxwell-Type Problems. In: Langer, U., Schanz, M., Steinbach, O., Wendland, W. (eds) Fast Boundary Element Methods in Engineering and Industrial Applications. Lecture Notes in Applied and Computational Mechanics, vol 63. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25670-7_1

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  • DOI: https://doi.org/10.1007/978-3-642-25670-7_1

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