An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 3, p. 875-902

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.

DOI : https://doi.org/10.1051/m2an/2013065
Classification:  65N30,  78A25
Keywords: Eddy current problems, time-dependent electromagnetic problems, input current intensities, voltage drops, finite elements
@article{M2AN_2013__47_3_875_0,
     author = {Berm\'udez, Alfredo and L\'opez-Rodr\'\i guez, Bibiana and Rodr\'\i guez, Rodolfo and Salgado, Pilar},
     title = {An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {3},
     year = {2013},
     pages = {875-902},
     doi = {10.1051/m2an/2013065},
     zbl = {1266.78025},
     mrnumber = {3056413},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_3_875_0}
}
Bermúdez, Alfredo; López-Rodríguez, Bibiana; Rodríguez, Rodolfo; Salgado, Pilar. An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 3, pp. 875-902. doi : 10.1051/m2an/2013065. http://www.numdam.org/item/M2AN_2013__47_3_875_0/

[1] R. Acevedo, S. Meddahi and R. Rodríguez, An E-based mixed formulation for a time-dependent eddy current problem. Math. Comput. 78 (2009) 1929-1949. | MR 2521273 | Zbl 1201.78027

[2] A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems. Numer. Methods Part. Differ. Equ. 21 (2005) 742-763. | MR 2140566 | Zbl 1079.78023

[3] A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems. IMA J. Numer. Anal. 24 (2004) 255-271. | MR 2046177 | Zbl 1114.78012

[4] A. Alonso and A. Valli, An optimal decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68 (1999) 607-631. | MR 1609607 | Zbl 1043.78554

[5] A. Alonso and A. Valli, Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications. Springer-Verlag, Italia (2010). | MR 2680968 | Zbl 1204.78001

[6] A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems. SIAM J. Appl. Math. 68 (2008) 1477-1494. | MR 2407134 | Zbl 1154.35454

[7] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. | MR 1626990 | Zbl 0914.35094

[8] A. Beranúdez, B. López-Rodríguez, R. Rodríguez and P. Salgado, Equivalence between two finite element methods for the eddy current problem. C. R. Math. Acad. Sci. Paris, Series I 34 (2010) 769-774. | Zbl 1201.78028

[9] A. Bermúdez, B. López-Rodríguez, R. Rodríguez and P. Salgado, Numerical solution of transient eddy current problems with input current intensities as boundary data. IMA J. Numer. Anal. 32 (2012) 1001-1029. | MR 2954738 | Zbl 1247.78035

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

[11] A. Bermúdez, R. Rodríguez and P. Salgado, Numerical analysis of electric field formulations of the eddy current model. Numer. Math. 102 (2005) 181-201. | Zbl 1084.78004

[12] A. Bossavit, Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998). | MR 1488417 | Zbl 0945.78001

[13] A. Bossavit, Most general non-local boundary conditions for the Maxwell equation in a bounded region. COMPEL 19 (2000) 239-245. | MR 1784010 | Zbl 0966.78002

[14] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl;Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845-876. | MR 1944792 | Zbl 1106.35304

[15] A. Buffa, Y. Maday and F. Rapetti, Applications of the mortar element method to 3D electromagnetic moving structures. Computational Electromagnetics, edited by C. Carstensen et al., Springer Verlag. Lect. Notes Comput. Sci. Eng. 28 (2003) 35-50. | MR 1986131 | Zbl 1065.78017

[16] C.R.I. Emson, and J. Simkin, An optimal method for 3D eddy currents. IEEE Trans. Magn. 19 (1983) 2450-2452.

[17] P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957-991. | MR 1479578 | Zbl 0910.35123

[18] P. Fernandes and I. Perugia, Vector potential formulation for magnetostatic and modelling of permanent magnets. IMA J. Appl. Math. 66 (2001) 293-318. | MR 1852930 | Zbl 0985.78005

[19] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[20] R. Hiptmair and O. Sterz, Current and voltage excitations for the eddy current model. Int. J. Numer. Model. 18 (2005) 1-21. | Zbl 1099.78021

[21] A. Kameari, Calculation of transient 3D eddy currents using edge elements. IEEE Trans. Magn. 26 (1990) 466-469.

[22] A. Kameari, Three dimensional eddy current calculation using edge elements for magnetic vector potential. Applied Electromagnetic in Materials, Pergamon Press, Oxford (1988) 225-236.

[23] T. Kang, T. Chen, H. Zhang and K.I. Kim, Improved T − ψ nodal finite element schemes for eddy current problems. Appl. Math. Comput. 218 (2011) 287-302. | MR 2820491 | Zbl 1227.78021

[24] C. Ma, The finite element analysis of a decoupled T-Ψ scheme for solving eddy-current problems. Appl. Math. Comput. 205 (2008) 352-361. | MR 2466639 | Zbl 1169.78005

[25] G. Pichenot, F. Buvat, V. Maillot and H. Voillaume, Eddy current modelling for non destructive testing. Proc. of 16th World Conf. on NDT, Rapport DSR 31. Montreal, August 30 - September 3 (2004).

[26] B. Weiß and O. Bíró, On the convergence of transient eddy-current problems. IEEE Trans. Magn. 40 (2004) 957-960.

[27] A. Žensíšek, Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. London, Academic Press (1990). | Zbl 0731.65090

[28] W. Zheng, Z. Chen and L. Wang, An adaptive finite element method for the H-ψ formulation of time-dependent eddy current problems. Numer. Math. 103 (2006) 667-689. | MR 2221067 | Zbl 1099.65092