Skip to main content
SpringerLink
Log in

A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper the discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods is established. We derive quasi-optimal a priori error estimates. Numerical examples prove the new scheme to be useful in practice. A posteriori error control and an adaptive algorithm are studied in Part II of this paper.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beck R, Hiptmair R, Hoppe RHW, Wohlmuth B (2000) Residual based a posteriori error estimators for eddy current computation. M2AN Math Model Numer Anal 34: 159–182

    Article  MATH  MathSciNet  Google Scholar 

  2. Bossavit A (1991) The computation of eddy-currents, in dimension 3, by using mixed finite elements and boundary elements in association. Math Comput Model 15: 33–42 Boundary integral equation methods (boundary element methods)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. In: Springer series in computational mathematics, vol 15. Springer-Verlag, New York

  4. Buffa A, Ciarlet P Jr (2001) On traces for functional spaces related to Maxwell’s equations. Part I: an integration by parts formula in Lipschitz polyhedra. Math Methods Appl Sci 24: 9–30

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Buffa A, Hiptmair R (2003) Galerkin boundary element methods for electromagnetic scattering. In: Topics in computational wave propagation. Lecturer Notes in Computer Science Engineering, vol 31. Springer, Berlin, pp 83–124

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Ciarlet PG (1978) The finite element method for elliptic problems. In: Studies in mathematics and its applications, vol 4. North-Holland, Amsterdam

  10. Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory. In: Applied mathematical sciences, vol 93, 2nd edn. Springer-Verlag, Berlin

  11. Costabel M (1987) Symmetric methods for the coupling of finite elements and boundary elements (invited contribution). In: Boundary elements IX, vol 1 (Stuttgart, 1987). Comput Mech, Southampton, pp 411–420

  12. Costabel M, Ervin VJ, Stephan EP (1990) Symmetric coupling of finite elements and boundary elements for a parabolic-elliptic interface problem. Quart Appl Math 48: 265–279

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Hiptmair R (2003) Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J Numer Anal 41: 919–944

    Article  MATH  MathSciNet  Google Scholar 

  16. MacCamy RC, Stephan E (1982) A simple layer potential method for three-dimensional eddy current problems. In: Ordinary and partial differential equations (Dundee, 1982). Lecture Notes in Mathematics, vol 964. Springer, Berlin, pp 477–484

  17. MacCamy RC, Stephan E (1983) A boundary element method for an exterior problem for three-dimensional Maxwell’s equations. Appl Anal 16: 141–163

    Article  MATH  MathSciNet  Google Scholar 

  18. MacCamy RC, Stephan E (1984) Solution procedures for three-dimensional eddy current problems. J Math Anal Appl 101: 348–379

    Article  MATH  MathSciNet  Google Scholar 

  19. MacCamy RC, Stephan E (1985) A skin effect approximation for eddy current problems. Arch Rational Mech Anal 90: 87–98

    Article  MATH  MathSciNet  Google Scholar 

  20. Mund P, Stephan EP (1997) Adaptive coupling and fast solution of FEM-BEM equations for parabolic-elliptic interface problems. Math Methods Appl Sci 20: 403–423

    Article  MATH  MathSciNet  Google Scholar 

  21. Nédélec J-C (1978) Computation of eddy currents on a surface in R 3 by finite element methods. SIAM J Numer Anal 15: 580–594

    Article  MATH  MathSciNet  Google Scholar 

  22. Nédélec J-C (1980) Mixed finite elements in R 3. Numer Math 35: 315–341

    Article  MATH  MathSciNet  Google Scholar 

  23. Nédélec J-C (1982) Integral equations with nonintegrable kernels. Integral Equ Operator Theory 5: 562–572

    Article  MATH  Google Scholar 

  24. Prato R (2009) FE/BE coupling for time-dependent interface problems in electromagnetics. PhD thesis, Institute of Applied Mathematics, Gottfried Wilhelm Leibniz Universität Hannover, Germany

  25. Stephan EP, Maischak M (2005) A posteriori error estimates for fem-bem couplings of three-dimensional electromagnetic problems. Comput Methods Appl Mech Eng 194: 441–452

    Article  MATH  MathSciNet  Google Scholar 

  26. Stephan EP, Maischak M, Leydecker F (2010) A residual error estimator for an electromagnetic fem-bem coupling problem in \({\mathbb{R}^3}\). Math Methods Appl Sci (to appear)

  27. Stephan EP, Maischak M, Leydecker F (2007) An hp-adaptive finite element/boundary element coupling method for electromagnetic problems. Comput Mech 39: 673–680

    Article  MATH  MathSciNet  Google Scholar 

  28. Stratton J, Chu L (1939) Diffraction theory of electromagnetic waves. Phys Rev II Ser 56: 99–107

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas y estadística, Universidad del Norte, Barranquilla, Colombia

    Ricardo A. Prato Torres

  2. Institut für Angewandte Mathematik, Gottfried Wilhelm Leibniz Universität Hannover, Hannover, Germany

    Ernst P. Stephan & Florian Leydecker

Authors
  1. Ricardo A. Prato Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ernst P. Stephan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Florian Leydecker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florian Leydecker.

Additional information

Communicated by C.C. Douglas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prato Torres, R.A., Stephan, E.P. & Leydecker, F. A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates. Computing 88, 131–154 (2010). https://doi.org/10.1007/s00607-010-0089-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-010-0089-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation