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.
Mots clés : Eddy current problem, corner singularity, edge singularity
@article{M2AN_2003__37_5_807_0, author = {Costabel, Martin and Dauge, Monique and Nicaise, Serge}, title = {Singularities of eddy current problems}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {807--831}, publisher = {EDP-Sciences}, volume = {37}, number = {5}, year = {2003}, doi = {10.1051/m2an:2003056}, mrnumber = {2020865}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2003056/} }
TY - JOUR AU - Costabel, Martin AU - Dauge, Monique AU - Nicaise, Serge TI - Singularities of eddy current problems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 807 EP - 831 VL - 37 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003056/ DO - 10.1051/m2an:2003056 LA - en ID - M2AN_2003__37_5_807_0 ER -
%0 Journal Article %A Costabel, Martin %A Dauge, Monique %A Nicaise, Serge %T Singularities of eddy current problems %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 807-831 %V 37 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003056/ %R 10.1051/m2an:2003056 %G en %F M2AN_2003__37_5_807_0
Costabel, Martin; Dauge, Monique; Nicaise, Serge. Singularities of eddy current problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 5, pp. 807-831. doi : 10.1051/m2an:2003056. http://www.numdam.org/articles/10.1051/m2an:2003056/
[1] A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg 143 (1997) 97-112. | Zbl
and ,[2] 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).
, and ,[3] A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60 (2000) 1805-1823. | Zbl
, and ,[4] Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. | Zbl
, , and ,[5] Resolution of the Maxwell equations in a domain with reentrant corners. RAIRO Modél. Math. Anal. Numér. 32 (1998) 359-389. | Numdam | Zbl
, and ,[6] Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159-182. | Numdam | Zbl
, , and ,[7] -theory of the Maxwell operator in arbitrary domains. Russian Math. Surveys 42 (1987) 75-96. | Zbl
and ,[8] 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. | Zbl
and ,[9] A singular field method for the solution of Maxwell's equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999) 2028-2044. | Zbl
, and ,[10] Two dual formulations of the 3D eddy-current problem. COMPEL 4 (1985) 103-116.
,[11] Electromagnétisme en vue de la modélisation. Springer-Verlag (1993). | MR | Zbl
,[12] Integral equation methods in scattering theory. John Wiley & Sons, Inc., New York, Pure Appl. Math. (1983). | MR | Zbl
and ,[13] 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). | EuDML | Numdam | Zbl
and ,[14] Stable asymptotics for elliptic systems on plane domains with corners. Comm. Partial Differential Equations 9 & 10 (1994) 1677-1726. | Zbl
and ,[15] Singularities of Maxwell's equations on polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221-276. | Zbl
and ,[16] Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements. Numer. Math. 93 (2002) 239-277. | Zbl
and ,[17] Singularities of Maxwell interface problems. ESAIM: M2AN 33 (1999) 627-649. | Numdam | Zbl
, and ,[18] Elliptic boundary value problems on corner domains. Springer-Verlag, Berlin L.N. in Math. 1341 (1988). | MR | Zbl
,[19] Numerical approximation of elliptic interface and corner problems. Habilitationsschrift, Bonn, Germany (1981).
,[20] Finite element methods for Navier-Stokes equations. Springer-Verlag, Springer Ser. Comput. Math. 5 (1986). | MR | Zbl
and ,[21] Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics. Pitman, Boston 24 (1985). | MR | Zbl
,[22] Symmetric coupling for eddy currents problems. SIAM J. Numer. Anal. 40 (2002) 41-65. | Zbl
,[23] Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227-313. | Zbl
,[24] Computation of singular solutions in elliptic problems and elasticity. RMA 5. Masson, Paris (1991). | Zbl
and ,[25] Minimal regularity of the solutions of some transmission problems. Math. Methods Appl. Sci. 26 (2003) 321-348. | Zbl
,[26] Polygonal interface problems. Peter Lang, Berlin (1993). | MR | Zbl
,[27] General interface problems I,II. Math. Methods Appl. Sci. 17 (1994) 395-450. | Zbl
and ,[28] Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation. Math. Methods Appl. Sci. 9 (1999) 855-898. | Zbl
and ,[29] Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784-816. | Zbl
,[30] On the boundary value problems of electro- and magnetostatics. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982) 165-174. | Zbl
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