Singularities of eddy current problems

Martin Costabel; Monique Dauge; Serge Nicaise[1]

  • [1] Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 5, page 807-831
  • ISSN: 0764-583X

Abstract

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We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11]. We describe the singularities of the solutions. They are related to edge and corner singularities of certain problems for the scalar Laplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.

How to cite

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Costabel, Martin, Dauge, Monique, and Nicaise, Serge. "Singularities of eddy current problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.5 (2003): 807-831. <http://eudml.org/doc/245491>.

@article{Costabel2003,
abstract = {We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11]. We describe the singularities of the solutions. They are related to edge and corner singularities of certain problems for the scalar Laplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.},
affiliation = {Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise},
author = {Costabel, Martin, Dauge, Monique, Nicaise, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Eddy current problem; corner singularity; edge singularity},
language = {eng},
number = {5},
pages = {807-831},
publisher = {EDP-Sciences},
title = {Singularities of eddy current problems},
url = {http://eudml.org/doc/245491},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Costabel, Martin
AU - Dauge, Monique
AU - Nicaise, Serge
TI - Singularities of eddy current problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 5
SP - 807
EP - 831
AB - We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11]. We describe the singularities of the solutions. They are related to edge and corner singularities of certain problems for the scalar Laplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.
LA - eng
KW - Eddy current problem; corner singularity; edge singularity
UR - http://eudml.org/doc/245491
ER -

References

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  1. [1] A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg 143 (1997) 97–112. Zbl0883.65096
  2. [2] A. Alonso-Rodriguez, F. Fernandes and A. Valli, Weak and strong formulations for the time-harmonic eddy-current problem in general domains. Report UTM. Dipartimento di Matematica, Univ. di Trento, Italy 603 (2001). Zbl1051.78004
  3. [3] H. Ammari, A. Buffa and J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60 (2000) 1805–1823. Zbl0978.35070
  4. [4] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. Zbl0914.35094
  5. [5] F. Assous, P. Ciarlet Jr. and E. Sonnendrucker, Resolution of the Maxwell equations in a domain with reentrant corners. RAIRO Modél. Math. Anal. Numér. 32 (1998) 359–389. Zbl0924.65111
  6. [6] R. Beck, R. Hiptmair, R.H.W. Hoppe and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159–182. Zbl0949.65113
  7. [7] M. Birman and M. Solomyak, L 2 -theory of the Maxwell operator in arbitrary domains. Russian Math. Surveys 42 (1987) 75–96. Zbl0653.35075
  8. [8] M. Birman and M. Solomyak, On the main singularities of the electric component of the electro-magnetic field in regions with screens. St. Petersburg. Math. J. 5 (1993) 125–139. Zbl0804.35127
  9. [9] A.-S. Bonnet-Ben Dhia, C. Hazard and S. Lohrengel, A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999) 2028–2044. Zbl0933.78007
  10. [10] A. Bossavit, Two dual formulations of the 3D eddy-current problem. COMPEL 4 (1985) 103–116. 
  11. [11] A. Bossavit, Electromagnétisme en vue de la modélisation. Springer-Verlag (1993). Zbl0787.65090MR1616583
  12. [12] D. Colton and R. Kress, Integral equation methods in scattering theory. John Wiley & Sons, Inc., New York, Pure Appl. Math. (1983). Zbl0522.35001MR700400
  13. [13] M. Costabel and M. Dauge, Singularités d’arêtes pour les problèmes aux limites elliptiques, in Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), Exp. No. IV, 12 p. École Polytech., Palaiseau (1992). Zbl0796.35033
  14. [14] M. Costabel and M. Dauge, Stable asymptotics for elliptic systems on plane domains with corners. Comm. Partial Differential Equations 9 & 10 (1994) 1677–1726. Zbl0814.35024
  15. [15] M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221–276. Zbl0968.35113
  16. [16] M. Costabel and M. Dauge, Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements. Numer. Math. 93 (2002) 239–277. Zbl1019.78009
  17. [17] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems. ESAIM: M2AN 33 (1999) 627–649. Zbl0937.78003
  18. [18] M. Dauge, Elliptic boundary value problems on corner domains. Springer-Verlag, Berlin L.N. in Math. 1341 (1988). Zbl0668.35001MR961439
  19. [19] M. Dobrowolski, Numerical approximation of elliptic interface and corner problems. Habilitationsschrift, Bonn, Germany (1981). 
  20. [20] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Springer Ser. Comput. Math. 5 (1986). Zbl0585.65077MR851383
  21. [21] P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics. Pitman, Boston 24 (1985). Zbl0695.35060MR775683
  22. [22] R. Hiptmair, Symmetric coupling for eddy currents problems. SIAM J. Numer. Anal. 40 (2002) 41–65. Zbl1010.78011
  23. [23] V.A. Kondrat’ev, Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227–313. Zbl0194.13405
  24. [24] D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. RMA 5. Masson, Paris (1991). Zbl0647.73010
  25. [25] D. Mercier, Minimal regularity of the solutions of some transmission problems. Math. Methods Appl. Sci. 26 (2003) 321–348. Zbl1035.35035
  26. [26] S. Nicaise, Polygonal interface problems. Peter Lang, Berlin (1993). Zbl0794.35040MR1236228
  27. [27] S. Nicaise and A.-M. Sändig, General interface problems I,II. Math. Methods Appl. Sci. 17 (1994) 395–450. Zbl0824.35014
  28. [28] S. Nicaise and A.-M. Sändig, Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation. Math. Methods Appl. Sci. 9 (1999) 855–898. Zbl0958.35023
  29. [29] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784–816. Zbl1001.65122
  30. [30] R. Picard, On the boundary value problems of electro- and magnetostatics. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982) 165–174. Zbl0516.35023

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