Title Page	4
	Copyright Page	5
	Preface	6
	Table of Contents	12
	1 Setting the problem	15
	1.1 Maxwell equations and time-harmonic Maxwell equations	15
	1.2 Eddy currents and eddy current approximation	18
	1.3 Geometrical setting and boundary conditions	22
	1.4 Harmonic fields in electromagnetism	24
	1.5 The complete eddy current model	29
	2 A mathematical justification of the eddy current model	34
	2.1 The E-based formulation of Maxwell equations	34
	2.2 The eddy current model as the low electric permittivity limit	38
	2.3 The eddy current model as the low-frequency limit	40
	2.3.1 Higher order convergence	43
	3 Existence and uniqueness of the solution	48
	3.1 Weak formulation, existence and uniqueness for the magnetic field	49
	3.2 Determination of the electric field	51
	3.3 Strong formulation for the magnetic field	56
	3.3.1 The Faraday equation for the “cutting” surfaces	59
	3.3.2 Suitability of other formulations	61
	3.4 Existence and uniqueness for the complete eddy current model	64
	3.5 Other boundary conditions	65
	4 Hybrid formulations for the electric and magnetic fields	71
	4.1 Hybrid formulation using the magnetic field in the insulator	72
	4.2 A saddle-point approach for the EC /HI formulation	74
	4.2.1 Finite element discretization	79
	4.3 A saddle-point approach for the H-based formulation	88
	4.4 Hybrid formulation using the electric field in the insulator	90
	4.5 A saddle-point approach for the HC /EI formulation	95
	4.5.1 Finite element discretization	99
	4.5.2 Some remarks on implementation	104
	4.5.3 Numerical results	109
	4.6 A saddle-point approach for the E-based formulation	116
	5 Formulations via scalar potentials	123
	5.1 The weak formulation in terms of HC and ?I	124
	5.2 The strong formulation in terms of HC and ?I	129
	5.2.1 A domain decomposition procedure	131
	5.3 The formulation in terms of EC and ??I	132
	5.3.1 A domain decomposition procedure	136
	5.4 Numerical approximation	137
	5.4.1 The determination of a vector potential for the density current Je,I	138
	5.4.2 Finite element approximation	140
	5.5 The finite element approximation of EI	152
	6 Formulations via vector potentials	158
	6.1 Formulation for the Coulomb gauge and its numerical approximation	159
	6.1.1 The weak formulation	165
	6.1.2 Existence and uniqueness of the solution to the weak formulation	172
	6.1.3 Numerical approximation	176
	6.1.4 Numerical results	181
	6.1.5 A penalized formulation for the electric field	188
	6.2 Formulation for the Lorenz gauge and its numerical approximation	191
	6.2.1 Decoupled weak formulations and alternative gauge conditions	194
	6.2.2 Well-posed formulations based on the Lorenz gauge	199
	6.2.3 Weak formulations and positiveness	202
	6.2.4 Numerical approximation	205
	6.3 Other potential formulations	206
	7 Coupled FEM–BEM approaches	216
	7.1 The (AC, VC ) ? ?I formulation	218
	7.2 The (AC, VC ) ? ?? weak formulation	220
	7.3 Existence and uniqueness of the weak solution	224
	7.4 Stability as ? goes to 0	227
	7.5 Numerical approximation	229
	7.5.1 The non-convex case	232
	7.6 Other FEM–BEM approaches	232
	7.6.1 The code TRIFOU	232
	7.6.2 An approach based on the magnetic field HC	235
	7.6.3 An approach based on the electric field EC	241
	8 Voltage and current intensity excitation	246
	8.1 The eddy current problem in the presence of electric ports	247
	8.1.1 Hybrid formulations in term of EC and ??I	249
	8.1.2 Formulations in terms of HC and ??I	259
	8.1.3 Formulations in terms of TC and ??I	261
	8.1.4 Finite element approximation	265
	8.1.5 Numerical results	269
	8.2 Voltage and current intensity excitation for an internal conductor	274
	8.2.1 Variational formulations	278
	9 Selected applications	286
	9.1 Metallurgical thermoelectrical problems	286
	9.1.1 Induction furnaces	287
	9.1.2 Metallurgical electrodes	290
	9.2 Bioelectromagnetism: EEG and MEG	297
	9.3 Magnetic levitation	304
	9.4 Power transformers	309
	9.5 Defect detection	314
	Appendix	319
	A.1 Functional spaces and notation	319
	A.2 Nodal and edge finite elements	323
	A.2.1 Grad-conforming finite elements	324
	A.2.2 Curl-conforming finite elements	327
	A.3 Orthogonal decomposition results	331
	A.3.1 First decomposition result	331
	A.3.2 Second decomposition result	334
	A.3.3 Third decomposition result	336
	A.4 More on harmonic fields	337
	References	340
	Index	353