A mixed-FEM and BEM coupling for a three-dimensional eddy current problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, p. 291-318

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

DOI : https://doi.org/10.1051/m2an:2003027
Classification:  65N30,  65N38,  65N15
Keywords: Eddy-current, boundary element, mixed finite element
@article{M2AN_2003__37_2_291_0,
     author = {Meddahi, Salim and Selgas, Virginia},
     title = {A mixed-FEM and BEM coupling for a three-dimensional eddy current problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
     pages = {291-318},
     doi = {10.1051/m2an:2003027},
     zbl = {1031.78012},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_2_291_0}
}
Meddahi, Salim; Selgas, Virginia. A mixed-FEM and BEM coupling for a three-dimensional eddy current problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, pp. 291-318. doi : 10.1051/m2an:2003027. http://www.numdam.org/item/M2AN_2003__37_2_291_0/

[1] A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999) 607-631. | Zbl 1043.78554

[2] 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. | Zbl 0978.35070

[3] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. | Zbl 0914.35094

[4] A. Bermúdez, R. Rodríguez and P. Salgado, A finite element method with Lagrange multipliers for low-frequency harmonic Maxell equations. SIAM J. Numer. Anal. 40 (2002) 1823-1849. | Zbl 1033.78009

[5] A. Bossavit and J. Vérité, The TRIFOU code: Solving the 3-D eddy-currents problem by using H as state variable. IEEE Trans. Mag. 19 (1983) 2465-2470.

[6] A. Bossavit, Two dual formulations of the 3-D eddy-currents problem. COMPEL 4 (1985) 103-116.

[7] A. Bossavit, A rationale for edge elements in 3-D field computations. IEEE Trans. Mag. 24 (1988) 74-79.

[8] A. Bossavit, 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. | Zbl 0725.65112

[9] A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris, Berlin, Heidelberg (1993). | MR 1616583 | Zbl 0787.65090

[10] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer, Berlin, Heidelberg, New York (1991). | MR 1115205 | Zbl 0788.73002

[11] A. Buffa, Hodge decompositions on the boundary of a polyhedron: the multi-connected case. Math. Models Methods Appl. Sci. 11 (2001) 1491-1504. | Zbl 1014.58002

[12] A. Buffa, Traces for functional spaces related to Maxwell equations: an overview, in Proceedings of GAMM-Workshop, Kiel (2001).

[13] A. Buffa and P. Ciarlet Jr., 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. | Zbl 0998.46012

[14] A. Buffa, M. Costabel and D. Sheen, On traces for 𝐇(rot,Ω) in Lipschitz domains. University of Pavia, IAN-CNR 1185 (2000). | Zbl 1106.35304

[15] 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. | Zbl 1019.65094

[16] 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). | MR 956899 | Zbl 0682.65069

[17] R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 5. Masson, Paris, Milan, Barcelone (1988). | MR 944303 | Zbl 0749.35005

[18] G.N. Gatica and G.C. Hsiao, Boundary-field equation methods for a class of nonlinear problems. Longman (1995). | MR 1379331 | Zbl 0832.65126

[19] V. Girault and P.A. Raviart, Finite element approximation of the Navier-Stokes equations: theory and algorithms. Springer, Berlin, Heidelberg, New York (1986). | MR 851383 | Zbl 0413.65081

[20] R. Hiptmair, Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40 (2002) 41-65. | Zbl 1010.78011

[21] C. Johnson and J.C. Nédélec, On the coupling of boundary integral and finite element methods. Math. Comp. 35 (1980) 1063-1079. | Zbl 0451.65083

[22] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press (2000). | MR 1742312 | Zbl 0948.35001

[23] S. Meddahi, An optimal iterative process for the Johnson-Nedelec method of coupling boundary and finite elements. SIAM J. Numer. Anal. 35 (1998) 1393-1415. | Zbl 0912.65096

[24] S. Meddahi, A mixed-FEM and BEM coupling for a two-dimensional eddy current problem. Numer. Funct. Anal. Optim. 22 (2001) 675-696. | Zbl 0996.78013

[25] S. Meddahi and F.J. Sayas, A fully discrete BEM-FEM for the exterior Stokes problem in the plane. SIAM J. Numer. Anal. 37 (2000) 2082-2102. | Zbl 0981.65129

[26] S. Meddahi, J. Valdés, O. Menéndez and P. Pérez, On the coupling of boundary integral and mixed finite element methods. J. Comput. Appl. Math. 69 (1996) 127-141. | Zbl 0854.65103

[27] J.C. Nédélec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[28] J.C. Nédélec, Acoustic and electromagnetic equations. Integral representations for harmonic problems. Springer-Verlag, New York (2001). | MR 1822275 | Zbl 0981.35002

[29] 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. | Zbl 0875.65090