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Trace Theorems on Non-Smooth Boundaries for Functional Spaces Related to Maxwell Equations: an Overview

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Computational Electromagnetics

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 28))

Summary

We study tangential vector fields on the boundary of a bounded Lipschitz domain in ℝ3. Our attention is focused on the definition of suitable Hilbert spaces over a range of Sobolev regularity which we try to make as large as possible, and also on the construction of tangential differential operators. Hodge decompositions are proved to hold for some special choices of spaces which are of interest in the theory of Maxwell equations.

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References

  1. C. Amrouche, C. Bernardi, M Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci., 21:823–864, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Buffa. Hodge decompositions on the boundary of a polyhedron: the multiconnected case. Math. Meth. Model. Appl. Sci., 11(9):1491–1504, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Buffa and S.A. Christiansen. The electric field integral equation on Lipschitz screens: definition and numerical approximation. Numer. Mathem., 2002 (in press)

    Google Scholar 

  4. 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., 21(1):9–30, 2001.

    Article  MathSciNet  Google Scholar 

  5. 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. Meth. Appl. Sci., 21(1):31–48, 2001.

    Article  MathSciNet  Google Scholar 

  6. A. Buffa, M. Costabel, and C. Schwab. Boundary element methods for Maxwell equations in non-smooth domains. Numer. Mathem., (electronic) DOI 10.1007/s002110100372, 2001.

    Google Scholar 

  7. A. Buffa, M. Costabel, and D. Sheen. On traces for H(curl, №) for Lipschitz domains. J. Math. Anal. Appl., 2002 (in press)

    Google Scholar 

  8. A. Buffa, R. Hipmair, T. von Petersdorff, and Ch. Schwab. Boundary element methods for Maxwell equations in Lipschitz domains. Numer. Mathem., 2002. (in press).

    Google Scholar 

  9. M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Meth. Applied Sei., 12:365–368, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Fernandes and G. Gilardi. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Mod. Meth. Appl. Sci., 7:957–991, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Fernandes and I. Perugia. Vector potential formulation for magnetostatics and modelling of permanent magnets. IMA J. of Appl. Mathem., 66:293–318, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Filonov. Systeme de Maxwell dans des domaines singuliers. PhD thesis, Université de Bordeaux 1, 1996.

    Google Scholar 

  13. N. Filonov. Principal singularities of the magnetic field in resonators with boundary of given smoothness. St. Petersburg Math. J., 9(2):379–390, 1998.

    MathSciNet  Google Scholar 

  14. V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations. Sringer-Verlag, Berlin, 1986.

    Book  MATH  Google Scholar 

  15. P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and studies in Mathematics. Pitman, London, 1985.

    Google Scholar 

  16. P. Grisvard. Singularities in boundary value problems, volume RMA 22. Masson, Paris, 1992.

    Google Scholar 

  17. R. Hiptmair. Symmetric coupling for eddy current problems. Technical Report 148, Sonderforschungsbereich 382, University of Tübigen, March 2000. Submitted to SIAM J. Numer. Anal.

    Google Scholar 

  18. D. Jerison and C.E. Kenig. The inhomogeneous dirichlet problem in lipschitz domains. J. Fund. Anal., 130:161–219, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Dunod, Paris, 1968.

    MATH  Google Scholar 

  20. Alan Mclntosh and Marius Mitrea. Clifford algebras and Maxwell’s equations in Lipschitz domains. Math. Methods Appl. Sci., 22(18):1599–1620, 1999.

    Article  MathSciNet  Google Scholar 

  21. M. Mitrea. Generalized Dirac operators on nonsmooth manifolds and Maxwell’s equations. J. Fourier Anal. Appl., 7(3):207–256, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Necăs. Les méthodes directes en théorie des équations elliptiques. Masson, Paris, 1967.

    MATH  Google Scholar 

  23. R. Picard. On the boundary value problem of electro-and magnetostatics. Proc. Royal Soc. Edinburgh, (92 A):165–174, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. H. Spanier. Algebraic Topology. McGraw Hill Book Company, 1966.

    Google Scholar 

  25. L. Tartar. On the characterization of traces of a sobolev space used for Maxwell’s equation. In Proceedings of a meeting held in Bordeaux, in honour of Michel Artola, November 1997.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Istituto di Matematica Applicata e Tecnologie Informatiche, Via Ferrata, 1, 27100, Pavia, Italy

    Annalisa Buffa

Authors
  1. Annalisa Buffa
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Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19717, USA

    Peter Monk

  2. Institute for Applied Mathematics, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria

    Carsten Carstensen

  3. Institute of Mathematics, University of Erlangen-Nürnberg, Bismarckstraße 1 1/2, 91054, Erlangen, Germany

    Stefan Funken

  4. Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Liepzig, Germany

    Wolfgang Hackbusch

  5. Department of Mathematics, University of Augsburg, Universitätsstraße 14, 86159, Augsburg, Germany

    Ronald H. W. Hoppe

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© 2003 Springer-Verlag Berlin Heidelberg

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Buffa, A. (2003). Trace Theorems on Non-Smooth Boundaries for Functional Spaces Related to Maxwell Equations: an Overview. In: Monk, P., Carstensen, C., Funken, S., Hackbusch, W., Hoppe, R.H.W. (eds) Computational Electromagnetics. Lecture Notes in Computational Science and Engineering, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55745-3_3

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  • DOI: https://doi.org/10.1007/978-3-642-55745-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44392-6

  • Online ISBN: 978-3-642-55745-3

  • eBook Packages: Springer Book Archive

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