SUBJECT INDEX
A
Abelian group, 282
Abraham-Minkowski debate, 18
abstract syntax, 278
accumulation point, 288
active nodes, 96, 98,
see also
free DoF
acute mesh, 83, 106, 110-119
acuteness, 112, 116-118
adherence, 288
adjoint, 315
admittance, 80
a.e. (almost everywhere), 308
affine relation, 286
affine space or subspace, 284-286
Aharonov-Bohm effect, 244, 274, 275
air mode, 253, 255, 259
algebra (of types), 276, 279, 323, 324
algebraic expression, 279
algebraic topology, 147, 148, 151, 152
altitude, 82
amp6re (unit), 3, 14
Amp6re rule of orientation, 5
Amp6re theorem, 3, 39, 40, 50, 135,
194, 235, 239
An, 283
analytic ellipticity, 254
analyticity, 254, 255
and
276
angle, 292
angular frequency, 219, 242, 251
anisotropy, 13, 14, 149
anti-linear function, 314
antisymmetric relation, 270
approximation error, 113, 117, 119,
171, 179, 288, 329, 336
approximation method, 71
arginf, 272
arithmetic expression, 281
Arrow-Hurwicz-Uzawa algorithm,
327
arrowed notation, 280
arrowed representation of fields, 301
artificial boundary, 49, 56, 200
assembly, 81, 82, 194
augmented Lagrangian, 327
augmented linear system, 173, 327
axial vector, 296, 297
B
b-oriented method, 54
Banach space, 66, 304
Banach theorem, 289
barrel, 292
barycentric coordinates, 75, 76, 286
barycentric map, 286
barycentric refinement, 99
basis functions, 71
Bath cube problem, 31, 50, 163, 195,
242, 302
Bath ladder, 303
Bean model, 266, 267
belted tree, 154, 156, 177, 235, 238
Beppo Levi space, 69, 197, 217, 226
Betti number, 146, 147, 148, 151, 160
bilateral estimate, xvii, 172, 176, 180,
189, 336
binding mechanisms, 278
bisymmetric matrix, 321
Biot and Savart formula, 23, 230, 237
boldface convention, 2-3, 78, 142, 270,
284, 320
Bolzano-Weierstrass proof, 290
Boolean algebra, 264, 276, 278
bound variable, 53, 307
boundary (of a simplex or chain), 151,
153, 154
boundary (topological), 288
boundary elements, xiii, 24, 193
bounding modulo ... (for a line or a
surface), 103
bound vector, 286
box, 100, 101, 102, 111
box method, 123,
see also
finite
volumes
Brillouin zones, 109
C
Cantor set, 307
cap (on a surface node), 98, 101, 102
capacitive effects, 222, 225
343
344 SUBJECT INDEX
Cartesian product, 316
Cauchy conditions, 88
Cauchy sequence, 304
Cauchy-Schwarz inequality, 310
canonical (isomorphism), 276
cell (of dual mesh),
see
dual cell
cgs system, 14
chain, 150
change of variables, 86, 90, 309
charge (electric), 3, 4, 8, 17, 24, 25
charge dynamics, 10, 11, 26
charge imbalance, 26
chart, 254, 287
Cholesky factorization, 321, 325
Chu formulation, 6
circuit, 291
circulation, 4, 40, 310
Clifford algebra, 1
closable operator, 126
closed chain, 154, 234
closed modulo... (for a surface or a
line), 102, 163
closed operator, 126, 316
closed set, closure, 288
cluster (of simplices), 75
clustering sequences, 288, 289, 290
co-edge, 152-156, 234-237, 335-336
coarseness of a mesh,
see
grain
cod, codomain, 265, 266, 320
coenergy, 48, 61-67, 122, 194-202
coercivity, 249, 315
cohomology, 133, 148
coil, 50, 193, 194, 220, 221, 303
collocation, 215
commutative diagram, 144, 296
compact operator, 251, 315
compact set, compact part, 42, 46, 251,
288
compactness (in existence argument),
71, 249, 290
compartment, 6-9, 15-17, 222, 248
compatibility of structures, 291
complementarity, xiii, xvii, 170-180,
189, 256, 329, 338
complementary trees, 335, 336
complementary subspace, 285
completion, 67, 304
complex conjugation, 219, 249
complexification, 219, 314
composition (of relations), 269
computer programming, 268,
see also
programming
concrete syntax, 278
condenser, 25
condition number, conditioning, 182,
327
conditional expression, 281
conductivity, 10, 17, 27, 39, 248
configuration space, 7, 8, 26, 27
conformal transformation, 84, 86, 88
connected set, 132
constitutive law, 1, 6, 9, 40, 135, 175,
248, 267
constrained linear system, 172-176,
199, 232, 233, 319, 320, 326, 330
constraints (on DoF vectors), 149,
172-178, 181, 331, 333, 334
constraints (on solutions), 63, 165, 169,
228
continuous function, 289
contractible space, 132, 299
contraction; 312
contravariant components, 295
convective derivative, 25
convergence of approximation, 79, 95,
111-119, 142
convergence of iterative process, 182,
186, 230 •
convex function, 168
convex hull, 107, 286
convex set, 286
convexity and mesh-generation, 107
convolution, 14, 216, 290
cotangent formula, 81, 91, 110, 118,
158
cotree, 125, 149-156, 186
coulomb (unit), 3
Coulomb force, 8, 11
Coulomb gauge, 134, 274, 275, 299
Coulomb gauge ("discrete"), 154
covariant components, 295
Crank-Nicolson method, 230
critical current, 267
critical point, 224, 227-229
cross product, 19-20, 141, 287,
294-296
crossing direction, 37-39, 137, 141, 287
curl, 295
curl-curl operator, xiv, xvii, 24, 182,
183, 239, 319
SUBJECT INDEX 345
current loop, 225, 235, 238
curvature, 69, 160, 209
curve, 309
curved tetrahedra, 74, 135, 200
cut, 58, 103, 163, 176, 178, 237-239,
329
cycle, 151-153
D
data structure, 74, 82, 135
de Rham's cohomology, 148
deformation-retract, 298
degree of freedom, 71, 78, 270
Delaunay (Delone, Deloney), 106,
see
also
Voronoi-Delaunay
dense part, 288
descent method, 182
diagonals (large and small), 86, 89
diamagnetic materials, 14
dielectric constant, 13
differential forms, 1, 40, 41, 55, 62,
153, 298
differential geometry, 41, 59, 125, 287,
295, 296
differential operators, 33, 41, 125, 132,
216, 239, 281, 294-296
differential operators (weak form of),
128, 197
dimension, 284
Dirac mass, 12, 216
direct frame, 287
directional derivative, 34, 63, 86, 87,
88
Dirichlet conditions, 55, 165
Dirichlet domains, 109
Dirichlet problem, 55, 202, 206, 208,
338
Dirichlet simplex, 108, 109,
see also
Voronoi-Delaunay
Dirichlet-to-Neumann map, 203-206,
211, 215, 240
discontinuity (at interface), 34, 36, 37,
48, 74, 110
discriminant, 62, 63, 168
disjoint union, 80
displacement current, 3, 20, 221, 247
distance, 288
distance in energy, 64
distributions (in the sense of), 12, 19,
41, 104, 130, 206, 217, 260, 275
div-conformal field, 97, 329, 338
divergence, 296
div-grad operator, xiv, xvii, 24, 239
dom, domain, 265, 266
domain (dual meaning of), 27, 33
domain (as open connected set), 3, 291
dominated convergence, 307
dot product, 292
double integrals, 214
double-multiplier approach, 328
doubling (of DoFs), 135, 178, 237
drift velocity, 10
dual cell, 100, 108, 139, 188
dual graph, 155, 336
dual mesh, 99, 100, 101, 108, 139
duality (of convex functions), 168
dualization of constraints, 59, 173
dummy variable, 264,
see also
bound
variable
dynamics, 5, 7-9, 11, 20, 195
E
E 3, 294
Earnshaw "theorem", 105
eddy currents, xv, 220
edge element, xv, 140
effective conditioning, 182-183
effective group action, 282
Eiffel tower, 300
eigenmode, 251
eigenvalue, 182, 183, 251, 252, 259, 261,
315,326
electret, 13, 15
electric potential, 22, 40
electromagnet, 193
electromagnetic energy, 16
electromagnetic field (as a mathemat-
ical structure vs as a physical
entity), 1, 5, 20, 32, 66, 274, 297
electromotive force, xv, 40, 243, 244
electron, 8, 9, 10, 13, 274, 275
electrostatics, 22, 23, 67, 105, 203, 211
eligible functions, 47, 52-54, 63, 65,
70, 164, 288
ellipsoid, 293
energy flux, 18
equation, 53
equivalence, 271
equivalence class, 68, 69, 88, 197,
271-274, 285, 287, 296, 308
346
SUBJECT INDEX
equivalent permeability, 21
error estimate, 112, 116-119, 132, 338
error estimator (for mesh refinement),
337
error in constitutive law, 168, 170, 175,
188, 198, 329, 330, 336
essential boundary conditions, 55, 165
Euclidean structure, 292, 293
Euler equation, 64, 71, 168, 169, 198,
199, 205, 207, 227, 229, 312
Euler-Poincar6 formula, 96, 99, 148,
184, 330, 334
Euratom casing, 303
exact sequence, 146, 299
expression, 279
extension (of a relation), 268
extension by continuity, 130, 305
external orientation, 58
extrusion, 36
F
f(x) (ambiguity of), 280
face element, 140
factorization,
see
LDL t, Cholesky,
Gauss
faithful group action, 282
false,
276
family, 269
fan map, 88
farad (unit), 6
Faraday's law, 3, 4, 5, 13, 36, 243, 274,
297, 298, 302
Fast Fourier Transform, 231
Felix brick 303
ferrite, 21
ferromagnetic materials, 14, 23
Feynman disk paradox, 20
finer than, 271
finite differences method, 85
finite volume methods, 102, 106, 111
finite volume of support, 197
fixed point condition, 9
flatness, 116-118, 183, 260
fluid dynamics, 25
flux, 4, 40, 310
flux loss, 96, 98, 101, 120, 121
for
all quantifier, 278
force, 8, 9, 19, 24, 25, 44, 60, 80, 194,
195, 297
Foucault currents, 220
Fourier component, 302
Fourier series, 304, 315
Fourier transform, 12, 14, 248
fractal, 67, 300
frame, 287
Fredholm alternative, 249, 315
free degrees of freedom, 80
free variable, 53, 264, 278-281
free vector, 286
Friedrichs inequality, 157,
see also
Poincar6 inequality
Fubini theorem, 132, 160, 216, 290, 308
functional, 62, 281
functional graph, 126, 129, 266, 316
functional point of view, 2-3, 52, 62,
65, 125, 164, 195
functional programming, 268
functional space, 64
G
Galerkin method, xvi, 71-73
Gateaux derivative, 63
gauge (w.r.t. a convex set), 293
gauge invariance, 18
gauging, 88, 154, 169, 174, 186, 274,
275, 299, 322, 333
gauss (unit), 14
Gauss factorization, 319
Gauss quadrature formulas, 78
Gauss-Seidel algorithm, 270
generalized sequence, 273
generalized solution, 70, 125-126
generators, 10, 15
gradient, 295
grain (of a mesh), 112-117, 142, 260,
337
graph (of a relation), 265
graph closing and hole plugging, 129
Green operator, 281
grot (for the
SCALAR ~ VECTOR
surface operator), 189
group, group action, 282
H
h-oriented method, 54
Hadamard, (well posed in the sense
of), 9,
see also
well-posed
problem
harmonic functions or fields, 35, 47,
84-88, 105, 144, 208, 209, 228,
242, 260
SUBJECT INDEX
347
harmonic time evolution, 21, 63, 219,
222, 223, 231, 247
hat function, 77
heat equation, 17
Heaviside notation for vectors, 3
Helmholtz decomposition, 166, 196,
197
henry (unit), 6
Hermitian elements, 78
Hermitian matrix, 230, 235
Hermitian scalar product, 219, 232,
249, 314
high permeability, 31, 39, 89, 92, 152
Hilbert problems, 35
Hilbert space, 66, 69, 125, 128, 310,
311
Hilbert space and dual, 207, 313
Hilbert transform, 248
Hilbertian geometry as extension of
Euclidean geometry, 312, 315
Hilbertian methods in BVP, 206
hole (as topological feature), 145, 147,
148, 154, 165, 225
holes (as charge carriers), 10
hollow tetrahedron, 147
holomorphy, 84, 88
homeomorphism, 87, 147, 148, 243,
289
homogeneity, 283, 293
homogeneity of physical space, 285
homogeneous space, 283
homology, 58, 102, 148-156
homology class, 164
homology cycle, 155, 156
homotopy, 291, 299
hypercircle, xvii, 169, 171, 189
hysteresis, 14, 21, 195, 267
I
i vs j (for x/-1)), 3
icosahedron, 335
image (of a set), 268
impedance, 224, 248
incidence matrice, incidence number,
75, 135-139, 143, 154, 158, 176,
201, 232-233
incindence matrices (for dual mesh),
139
incidence number, 136, 137, 138, 143,
176
independence (in linear space), 284
induction heating, 17, 21, 220, 221, 267
inertia, 28, 194
infimum, supremum, 272
infinite dimensional space, 284
infinite elements, 201
inf-sup condition, 161
inner orientation, 137, 138, 287, 295
integration by parts, 16, 25, 42, 45,
65, 129, 187, 202, 206, 211, 224,
253, 309, 310, 315
integral method, 202
integration-by-parts formulas, 130
interaction (between compartments),
1, 6, 9, 13-17, 275
interface condition, 44,
see also
discontinuity
interference, 274, 275
interior (of a set), 288
interpolation error, 113-117, 119
inverse (of a relation), 266
inverse image, 268
inverse problem, 23
involution, 302
isometry, 289
isomorphism, 276
isosceles tetrahedron, 85
isotropy, 283, 293
isotropy group, 283, 293,
see also
little
group
J
Jacobi identity, 118
Jacobian, 27, 90
Jamet criterion, 117
Joule heating, Joule losses, 15, 17, 165
jump, 34-39, 96-98, 102, 181, 187, 210,
238, 241
K
ker, 320
kernel, 286
kinematics, 5, 7
Kirchhoff laws, xv
knot, 234, 239-244
L
L 2, 308
L 6, 69, 197
L ~, 93
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