Contents
Preface xiii
CHAPTER 1 Introduction: Maxwell Equations
1.1 Field Equations 1
1.2 Constitutive Laws 6
1.2.1 Dynamics of free charges: the Vlasov-Maxwell model 7
1.2.2 Dynamics of conduction charges: Ohm's law 9
1.2.3 Dynamics of bound charges: dielectric polarization 11
1.2.4 Magnetization 13
1.2.5 Summing up: linear materials 14
1.3 Macroscopic Interactions 15
1.3.1 Energy balance 16
1.3.2 Momentum balance 18
1.4 Derived Models 20
Exercises 24
Solutions 26
References 28
CHAPTER2 Magnetostatics: "Scalar Potential" Approach 31
2.1 Introduction: A Model Problem
2.2 Honing Our Tools
2.2.1 Regularity and discontinuity of fields
2.2.2 Jumps
2.2.3 Alternatives to the standard formalism
2.3 Weak Formulations
2.3.1 The "divergence" side
2.3.2 The "curl" side
2.3.3 The uniqueness issue
31
33
33
37
39
41
42
45
47
vii
viii
CONTENTS
2.4 Modelling: The Scalar Potential Formulation
2.4.1 Restriction to a bounded domain
2.4.2 Introduction of a magnetic potential
2.4.3 Uniqueness
2.4.4 Laplace, Poisson, Dirichlet, and Neumann
Exercises
Solutions
References
48
48
51
53
55
56
57
59
CHAPTER 3 Solving for the Scalar Magnetic Potential
3.1 The "Variational" Formulation
3.2 Existence of a Solution
3.2.1 Trying to find one
3.2.2 ~* is too small
3.2.3 Completing ~*
3.3 Discretization
3.3.1 The Ritz-Galerkin method
3.3.2 Finite elements
3.3.3 The linear system
3.3.4 "Assembly", matrix properties
Exercises
Solutions
References
61
61
65
65
67
68
70
71
74
78
81
84
87
93
CHAPTER 4 The Approximate Scalar Potential: Properties
and Shortcomings
4.1 The "m-weak" Properties
4.1.1 Flux losses
4.1.2 The dual mesh, and which fluxes are conserved
4.1.3 The flux through S"
4.2 The Maximum Principle
4.2.1 Discrete maximum principle
4.2.2 Voronoi-Delaunay tessellations and meshes
4.2.3 VD meshes and Stieltjes matrices
4.3 Convergence and Error Analysis
4.3.1 Interpolation error and approximation error
4.3.2 Taming the interpolation error: Zlamal's condition
4.3.3 Taming the interpolation error: flatness
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95
96
99
103
105
105
107
109
111
113
114
116
CONTENTS
ix
Exercises
Solutions
References
119
120
123
CHAPTER 5 Whitney Elements
5.1 A Functional Framework
5.1.1 The "weak" grad, rot, and div
5.1.2 New functional spaces: L 2 IL 2 IL 2
grad / rot I div
5.1.3 Proof of Proposition 5.2
5.1.4 Extending the Poincar6 lemma
5.1.5 "Maxwell's house"
5.2 The Whitney Complex
5.2.1 Oriented simplices
5.2.2 Whitney elements
5.2.3 Combinatorial properties of the complex
5.2.4 Topological properties of the complex
5.2.5 Metric properties of the complex
5.3 Trees and Cotrees
5.3.1 Homology
5.3.2 Trees, co-edges
5.3.3 Trees and graphs
Exercises
Solutions
References
125
125
126
128
131
132
134
135
136
139
142
145
148
149
150
152
155
157
158
161
CHAPTER 6 The "Curl Side": Complementarity
163
6.1 A Symmetrical Variational Formulation
6.1.1 Spaces of admissible fields
6.1.2 Variational characterization of the solution
6.1.3 Complementarity, hypercircle
6.1.4 Constrained linear systems
6.2 Solving the Magnetostatics Problem
164
164
167
169
172
174
6.2.1 Embedding the problem in Maxwell-Whitney's house 175
6.2.2 Dealing with the constraints 178
6.2.3 "m-weak" properties 179
6.3 Why Not Standard Elements? 180
6.3.1 An apparent advantage:
Mp
is regular 180
6.3.2 Accuracy is downgraded 181
x CONTENTS
6.3.3 The "effective" conditioning of the final matrix is
worsened
6.3.4 Yes, but...
6.3.5 Conclusion
Exercises
Solutions
References
182
183
185
187
188
189
CHAPTER 7 Infinite Domains
7.1 Another Model Problem
7.2 Formulation
7.2.1 Functional spaces
7.2.2 Variational formulations
7.3 Discretization
7.3.1 First method: "artificial boundary, at a distance"
7.3.2 Second method: "infinite elements"
7.3.3 Third method: "finite elements and integral method
in association"
7.4 The "Dirichlet-to-Neumann" Map
7.4.1 The functional space of "traces"
7.4.2 The interior Dirichlet-to-Neumann map
7.4.3 The exterior Dirichlet-to-Neumann map
7.4.4 Integral representation ....
7.4.5 Discretization
7.5 Back to Magnetostatics
Exercises
Solutions
References
193
193
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196
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201
202
203
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205
207
208
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217
CHAPTER 8 Eddy-Current Problems
8.1 The Model in h
8.1.1 A typical problem
8.1.2 Dropping the displacement-currents term
8.1.3 The problem in h, inthe harmonic regime
8.2 Infinite Domains: "Trifou"
8.2.1 Reduction to a problem on C
8.2.2 The space H~, isomorphic to IK g
8.2.3 Reformulation of the problem in H~
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225
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