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A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

Published online by Cambridge University Press:  15 November 2003

Salim Meddahi  and
Virginia Selgas
Salim Meddahi
Affiliation:
Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, España. salim@orion.ciencias.uniovi.es., selgas@orion.ciencias.uniovi.es.
Virginia Selgas
Affiliation:
Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, España. salim@orion.ciencias.uniovi.es., selgas@orion.ciencias.uniovi.es.
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Abstract

We study in this paper the electromagnetic field generated in aconductor by an alternating current density. The resultinginterface problem (see Bossavit (1993)) between the metal and thedielectric medium is treated by a mixed–FEM and BEM couplingmethod. We prove that our BEM-FEM formulation is well posed andthat it leads to a convergent Galerkin method.

Keywords

Eddy–currentboundary elementmixed finite element.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 291 - 318
Copyright
© EDP Sciences, SMAI, 2003

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References

A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999) 607-631.
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
Bermúdez, A., Rodríguez, R. and Salgado, P., A finite element method with Lagrange multipliers for low-frequency harmonic Maxell equations. SIAM J. Numer. Anal. 40 (2002) 1823-1849. CrossRef
Bossavit, A. and Vérité, J., The TRIFOU code: Solving the 3-D eddy-currents problem by using H as state variable. IEEE Trans. Mag. 19 (1983) 2465-2470. CrossRef
Bossavit, A., Two dual formulations of the 3-D eddy-currents problem. COMPEL 4 (1985) 103-116. CrossRef
Bossavit, A., A rationale for edge elements in 3-D field computations. IEEE Trans. Mag. 24 (1988) 74-79. CrossRef
Bossavit, A., The computation of eddy-currents in dimension 3 by using mixed finite elements and boundary elements in association. Math. Comput. Modelling 15 (1991) 33-42. CrossRef
A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris, Berlin, Heidelberg (1993).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer, Berlin, Heidelberg, New York (1991).
Buffa, A., Hodge decompositions on the boundary of a polyhedron: the multi-connected case. Math. Models Methods Appl. Sci. 11 (2001) 1491-1504. CrossRef
A. Buffa, Traces for functional spaces related to Maxwell equations: an overview, in Proceedings of GAMM-Workshop, Kiel (2001).
Buffa, A. and Ciarlet Jr, P.., On traces for functional spaces related to Maxwell's equation. Part I: An integration by parts formula in lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9-30. 3.0.CO;2-2>CrossRef
A. Buffa, M. Costabel and D. Sheen, On traces for H(curl,Ω) in Lipschitz domains. University of Pavia, IAN-CNR 1185 (2000).
A. Buffa, M. Costabel and Ch. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains. Numer. Math. (2001) (electronic) DOI 10.1007/s002110100372.
M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements, in The Mathematics of Finite Elements and Applications IV, Academic Press, London (1988).
R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 5. Masson, Paris, Milan, Barcelone (1988).
G.N. Gatica and G.C. Hsiao, Boundary-field equation methods for a class of nonlinear problems. Longman (1995).
V. Girault and P.A. Raviart, Finite element approximation of the Navier-Stokes equations: theory and algorithms. Springer, Berlin, Heidelberg, New York (1986).
Hiptmair, R., Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40 (2002) 41-65. CrossRef
Johnson, C. and Nédélec, J.C., On the coupling of boundary integral and finite element methods. Math. Comp. 35 (1980) 1063-1079. CrossRef
W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press (2000).
Meddahi, S., An optimal iterative process for the Johnson-Nedelec method of coupling boundary and finite elements. SIAM J. Numer. Anal. 35 (1998) 1393-1415. CrossRef
Meddahi, S., A mixed-FEM and BEM coupling for a two-dimensional eddy current problem. Numer. Funct. Anal. Optim. 22 (2001) 675-696. CrossRef
Meddahi, S. and Sayas, F.J., A fully discrete BEM-FEM for the exterior Stokes problem in the plane. SIAM J. Numer. Anal. 37 (2000) 2082-2102. CrossRef
Meddahi, S., Valdés, J., Menéndez, O. and Pérez, P., On the coupling of boundary integral and mixed finite element methods. J. Comput. Appl. Math. 69 (1996) 127-141. CrossRef
Nédélec, J.C., Mixed finite elements in $\xR^3$ . Numer. Math. 35 (1980) 315-341. CrossRef
J.C. Nédélec, Acoustic and electromagnetic equations. Integral representations for harmonic problems. Springer-Verlag, New York (2001).
J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1991) 523-639.