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Singularities of eddy current problems

Published online by Cambridge University Press:  15 November 2003

Martin Costabel ,
Monique Dauge  and
Serge Nicaise
Martin Costabel
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. ,
Monique Dauge
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. ,
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France.
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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.

Keywords

Eddy current problemcorner singularityedge singularity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 5 , September 2003 , pp. 807 - 831
Copyright
© EDP Sciences, SMAI, 2003

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References

Alonso, A. and Valli, A., A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg 143 (1997) 97-112. CrossRef
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).
Ammari, H., Buffa, A. and Nédélec, J.-C., A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60 (2000) 1805-1823.
Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. 3.0.CO;2-B>CrossRef
Assous, F., Ciarlet Jr, P.. and E. Sonnendrucker, Resolution of the Maxwell equations in a domain with reentrant corners. RAIRO Modél. Math. Anal. Numér. 32 (1998) 359-389. CrossRef
Beck, R., Hiptmair, R., Hoppe, R.H.W. and Wohlmuth, B., Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159-182. CrossRef
Birman, M. and Solomyak, M., L 2-theory of the Maxwell operator in arbitrary domains. Russian Math. Surveys 42 (1987) 75-96. CrossRef
Birman, M. and Solomyak, M., 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.
Bonnet-Ben Dhia, A.-S., Hazard, C. and Lohrengel, S., A singular field method for the solution of Maxwell's equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999) 2028-2044. CrossRef
Bossavit, A., Two dual formulations of the 3D eddy-current problem. COMPEL 4 (1985) 103-116. CrossRef
A. Bossavit, Electromagnétisme en vue de la modélisation. Springer-Verlag (1993).
D. Colton and R. Kress, Integral equation methods in scattering theory. John Wiley & Sons, Inc., New York, Pure Appl. Math. (1983).
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).
M. Costabel and M. Dauge, Stable asymptotics for elliptic systems on plane domains with corners. Comm. Partial Differential Equations 9 & 10 (1994) 1677-1726.
Costabel, M. and Dauge, M., Singularities of Maxwell's equations on polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221-276. CrossRef
Costabel, M. and Dauge, M., Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements. Numer. Math. 93 (2002) 239-277. CrossRef
Costabel, M., Dauge, M. and Nicaise, S., Singularities of Maxwell interface problems. ESAIM: M2AN 33 (1999) 627-649. CrossRef
M. Dauge, Elliptic boundary value problems on corner domains. Springer-Verlag, Berlin L.N. in Math. 1341 (1988).
M. Dobrowolski, Numerical approximation of elliptic interface and corner problems. Habilitationsschrift, Bonn, Germany (1981).
V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Springer Ser. Comput. Math. 5 (1986).
P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics. Pitman, Boston 24 (1985).
Hiptmair, R., Symmetric coupling for eddy currents problems. SIAM J. Numer. Anal. 40 (2002) 41-65. CrossRef
Kondrat'ev, V.A., Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227-313.
D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. RMA 5. Masson, Paris (1991).
Mercier, D., Minimal regularity of the solutions of some transmission problems. Math. Methods Appl. Sci. 26 (2003) 321-348. CrossRef
S. Nicaise, Polygonal interface problems. Peter Lang, Berlin (1993).
Nicaise, S. and Sändig, A.-M., General interface problems I,II. Math. Methods Appl. Sci. 17 (1994) 395-450. CrossRef
Nicaise, S. and Sändig, A.-M., Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation. Math. Methods Appl. Sci. 9 (1999) 855-898. CrossRef
Nicaise, S., Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784-816. CrossRef
Picard, R., On the boundary value problems of electro- and magnetostatics. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982) 165-174. CrossRef