Abstract. The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain.Then, the curl-free condition is imposed, at a discrete level, by introducing a piecewise linear multivalued potential. The resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential. Moreover, this approach leads to an important saving in computational effort. Problems related to the topology are also considered in that the possibility of having a non simply connected dielectric domain is taken into account.Implementation issues are discussed, including an amenable procedure to impose the boundary conditions by means of a Lagrange multiplier. Finally the method is applied to solve a threedimensional model problem: a cylindrical electrode surrounded by dielectric.Key words. low-frequency harmonic Maxwell equations, eddy currents problems, finite element computational electromagnetism

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Abstract. The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.Résumé. L'objectif de cet article est d'analyser une formulation du problème des courants de Foucault,écrite en fonction d'une primitive en temps du champélectrique, dans un domaine borné, et etant les intensités du courant ou les chutes de potentiel les sources données du problème.À ce propos, nous introduisons un multiplicateur de Lagrange pour imposer la condition de divergence nulle dans le domain diélectrique et nous obtenons une formulation faible mixte conduisantà un problème parabolique dégénéré, pour lequel l'existence et l'unicité d'une solution sont démontrées. Nous proposons une discretisation spatiale du problème, basée sur deséléments finis d'arête de Nédélec pour la variable principale et sur deséléments finis nodaux standard pour le multiplicateur de Lagrange. Nous obtenons des estimations d'erreur pour cette discrétisation. Nous introduisons ensuite un schéma d'Euler implicite pour la discretisation en temps et nous démontrons des estimations d'erreur por le problème complètement discretisé. Finalement, la méthode est appliquéeà la résolution de quelques problèmes test.1991 Mathematics Subject Classification. 65N30, 78A25.

Abstract. This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a physical point of view. Two different cases are considered depending on the boundary data: input current density flux or input current intensities. The analysis of the former is an intermediate step for the latter, which is more realistic in actual applications. Weak formulations in terms of the magnetic field are studied, the boundary conditions being imposed by means of appropriate Lagrange multipliers. The resulting mixed formulations are analyzed depending on the regularity of the boundary data. Finite element methods are introduced in each case and error estimates are proved. Finally, some numerical results to assess the effectiveness of the methods are reported.

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