<< . .

. 58
( : 59)



. . >>

CG, 223
multigrid iteration, 243
nested iteration, 253 Banach space, 404
Newton™s method, 357 Banach™s ¬xed-point theorem, 345
algorithmic error, 200 barycentric coordinates, 117
angle condition, 173 basis of eigenvalues
angle criterion, 184 orthogonal, 300
anisotropic, 8, 139 best approximation error, 70
ansatz space, 56, 67 BICGSTAB method, 238
nested, 240 bifurcation, 363
properties, 67 biharmonic equation, 111
approximation bilinear form, 400
superconvergent, 193 bounded, 403
approximation error estimate, 139, continuous, 93
144 de¬nite, 400
for quadrature rules, 160 positive, 400
one-dimensional, 137 positive de¬nite, 400
approximation property, 250 symmetric, 400
aquifer, 7 V -elliptic, 93
Armijo™s rule, 357 Vh -elliptic, 156
Arnoldi™s method, 235 block-Gauss“Seidel method, 211
algorithm, 235 block-Jacobi method, 211
416 Index

Bochner integral, 289 conjugate gradient, see CG
boundary, 393 connectivity condition, 173
boundary condition, 15 conormal, 16
Dirichlet, 15 conormal derivative, 98
¬‚ux, 15 conservative form, 14
homogeneous, 15 conservativity
inhomogeneous, 15 discrete global, 278
mixed, 15 consistency, 28
Neumann, 16 consistency error, 28, 156
boundary point, 393 constitutive relationship, 7
boundary value problem, 15 continuation method, 357, 363
adjoint, 145 continuity, 402
regular, 145 continuous problem, 21
weak solution, 107 approximation, 21
Bramble“Hilbert lemma, 135 contraction, 402
bulk density, 12 contraction number, 199
control domain, 257
Cantor™s function, 53 control volume, 257
capillary pressure, 10 convection
Cauchy sequence, 404 forced, 5, 12
Cauchy“Schwarz inequality, 400 natural, 5
CG method, 221 convection-di¬usion equation, 12
algorithm, 223 convection-dominated, 268
error reduction, 224 convective part, 12
with preconditioning, 228 convergence, 27
CGNE method, 235 global, 343
CGNR method, 234 linear, 343
characteristics, 388 local, 343
Chebyshev polynomial, 225 quadratic, 343
Cholesky decomposition, 84 superlinear, 343
incomplete, 231 with order of convergence p, 343
modi¬ed with respect to a norm, 401
incomplete, 232 correction, 201
chord method, 354 Crank-Nicolson method, 313
circle criterion, 184 cut-o¬ strategy, 187
closure, 393 Cuthill“McKee method, 89
coarse grid correction, 242, 243
coe¬cient, 16 Darcy velocity, 7
collocation method, 68 Darcy™s law, 8
collocation point, 68 decomposition
column sum criterion regular, 232
strict, 398 de¬niteness, 400
comparison principle, 40, 328 degree of freedom, 62, 115, 120
completion, 404 Delaunay triangulation, 178, 263
complexity, 88 dense, 96, 288, 404
component, 5 density, 7
condition number, 209, 397 derivative
spectral, 398 generalized, 53
conjugate, 219 material, 388
Index 417

weak, 53, 289 duality argument, 145
diagonal ¬eld, 362
diagonal scaling, 230 edge swap, 181
diagonal swap, 181 eigenfunction, 285
di¬erence quotient, 23 eigenvalue, 285, 291, 394
backward, 23 eigenvector, 291, 394
forward, 23 element, 57
symmetric, 23 isoparametric, 122, 169
di¬erential equation element sti¬ness matrix, 78
convection-dominated, 12, 368 element-node table, 74
degenerate, 9 ellipticity
elliptic, 17 uniform, 100
homogeneous, 16 embedding, 403
H k („¦) in C(„¦), 99
¯
hyperbolic, 17
inhomogeneous, 16 empty sphere criterion, 178
linear, 16 energy norm, 218
nonlinear, 16 energy norm estimates, 132
order, 16 energy scalar product, 217
parabolic, 17 equidistribution strategy, 187
quasilinear, 16 error, 201
semilinear, 16, 360 error equation, 68, 242
type of, 17 error estimate
di¬erential equation model a priori, 131, 185
instationary, 8 anisotropic, 144
linear, 8 error estimator
stationary, 8 a posteriori, 186
di¬usion, 5 asymptotically exact, 187
di¬usive mass ¬‚ux, 11 e¬cient, 186
di¬usive part, 12 reliable, 186
Dirichlet domain, 262 residual, 188
Dirichlet problem dual-weighted, 194
solvability, 104 robust, 187
discrete problem, 21 error level
discretization, 21 relative, 199
¬ve-point stencil, 24 Euler method
upwind, 372 explicit, 313
discretization approach, 55 implicit, 313
discretization parameter, 21 extensive quantity, 7
divergence, 20 extrapolation factor, 215
divergence form, 14 extrapolation method, 215
domain, 19, 394
C l , 407 face, 123
C k -, 96 family of triangulations
C ∞ -, 96 quasi-uniform, 165
Lipschitz, 96, 407 regular, 138
strongly, 407 Fick™s law, 11
domain of (absolute) stability, 317 ¬ll-in, 85
Donald diagram, 265 ¬nite di¬erence method, 17, 24
dual problem, 194 ¬nite element, 115, 116
418 Index

C 1 -, 115, 127 Lebesgue integrable, 407
a¬ne equivalent, 122 measurable, 393
Bogner“Fox“Schmit rectangle, 127 piecewise continuous, 48
C 0 -, 115 support, 394
cubic ansatz on simplex, 121 functional, 403
cubic Hermite ansatz on simplex, functional matrix, 348
126 functions
d-polynomial ansatz on cuboid, 123 equal almost everywhere, 393
equivalent, 122
Hermite, 126 Galerkin method, 56
Lagrange, 115, 126 stability, 69
linear, 57 unique solvability, 63
linear ansatz on simplex, 119 Galerkin product, 248
quadratic ansatz on simplex, 120 Galerkin/least squares“FEM, 377
simplicial, 117 Gauss™s divergence theorem, 14, 47,
¬nite element code 266
assembling, 176 Gauss“Seidel method, 204
kernel, 176 convergence, 204, 205
post-processor, 176 symmetric, 211
¬nite element discretization Gaussian elimination, 82
conforming, 114 generating function, 316
condition, 115 GMRES method, 235
nonconforming, 114 truncated, 238
¬nite element method, 18 with restart, 238
characterization, 67 gradient, 20
convergence rate, 131 gradient method, 218
maximum principle, 175 error reduction, 219
mortar, 180 gradient recovery, 192
¬nite volume method, 18 graph
cell-centred, 258 dual, 263
cell-vertex, 258 grid
node-centred, 258 chimera, 180
semidiscrete, 297 combined, 180
¬ve-point stencil, 24 hierarchically structured, 180
¬xed point, 342 logically structured, 177
¬xed-point iteration, 200, 344 overset, 180
consistent, 200 structured, 176
convergence theorem, 201 in the strict sense, 176
¬‚uid, 5 in the wider sense, 177
Fourier coe¬cient, 287 unstructured, 177
Fourier expansion, 287 grid adaptation, 187
Friedrichs“Keller triangulation, 64 grid coarsening, 183
frontal method, 87 grid function, 24
full discretization, 293 grid point, 21, 22
full upwind method, 373 close to the boundary, 24, 327
function far from the boundary, 24, 327
almost everywhere vanishing, 393 neighbour, 23
continuous, 407
essentially bounded, 405 harmonic, 31
Index 419

heat equation, 9 L0 -matrix, 399
Hermite element, 126 L-matrix, 399
Hessenberg matrix, 398 Lagrange element, 115, 126
Hilbert space, 404 Lagrange“Galerkin method, 387
homogenization, 6 Lagrangian coordinate, 387
hydraulic conductivity, 8 Lanczos biorthogonalization, 238
Langmuir model, 12
IC factorization, 231 Laplace equation, 9
ill-posedness, 16 Laplace operator, 20
ILU factorization, 231 lemma
existence, 232 Bramble“Hilbert, 135
ILU iteration, 231 C´a™s, 70
e
inequality ¬rst of Strang, 155
of Kantorovich, 218 lexicographic, 25
Friedrichs™, 105 linear convergence, 199
inverse, 376 Lipschitz constant, 402
of Poincar´, 71
e Lipschitz continuity, 402
in¬‚ow boundary, 108 load vector, 62
inhomogeneity, 15 LU factorization, 82
initial condition, 15 incomplete, 231
initial-boundary value problem, 15
inner product M-matrix, 41, 399
on H 1 („¦), 54 macroscale, 6
integral form, 14 mapping
integration by parts, 97 bounded, 402
interior, 394 continuous, 402
interpolation contractive, 402
local, 58 linear, 402
interpolation error estimate, 138, 144 Lipschitz continuous, 402
one-dimensional , 136 mass action law, 11
interpolation operator, 132 mass average mixture velocity, 7
interpolation problem mass lumping, 314, 365
local, 120 mass matrix, 163, 296, 298
isotropic, 8 mass source density, 7
iteration matrix
inner, 355 band, 84
outer, 355 bandwidth, 84
iteration matrix, 200 consistently ordered, 213
iterative method, 342 Hessenberg, 398
hull, 84
Jacobi matrix, 348 inverse monotone, 41
Jacobi™s method, 203 irreducible, 399
convergence, 204, 205 L0 -, 399
jump, 189 L-, 399
jump condition, 14 LU factorizable, 82
M-, 399
Krylov (sub)space, 222 monotone, 399
Krylov subspace of monotone type, 399
method, 223, 233 pattern, 231
420 Index

positive de¬nite, 394 ¬nite di¬erence, 24
pro¬le, 84 full upwind, 373
reducible, 399 Galerkin, 56
row bandwidth, 84 Gauss“Seidel, 204
row diagonally dominant GMRES, 235
strictly, 398 iterative, 342
weakly, 399 Jacobi™s, 203
sparse, 25, 82, 198 Krylov subspace, 223, 233
symmetric, 394 Lagrange“Galerkin, 387
triangular linear stationary, 200
lower, 398 mehrstellen, 30
upper, 398 moving front, 179
matrix norm multiblock, 180
compatible, 396 multigrid, 243
induced, 397 Newton™s, 349
mutually consistent, 396 of bisection, 182
submultiplicative, 396 stage number of, 182
subordinate, 397 one-step, 316
matrix polynomial, 394 one-step-theta, 312
matrix-dependent, 248 overlay, 177
max-min-angle property, 179 PCG, 228, 229
maximum angle condition, 144 r-, 181
maximum column sum, 396 relaxation, 207
maximum principle Richardson, 206
strong, 36, 39, 329 Ritz, 56
weak, 36, 39, 329 Rothe™s, 294
maximum row sum, 396 semi-iterative, 215
mechanical dispersion, 11 SOR, 210
mesh width, 21 SSOR, 211
method streamline upwind Petrov“
advancing front, 179, 180 Galerkin, 375
algebraic multigrid, 240 streamline-di¬usion, 377
Arnoldi™s , 235 method of conjugate directions, 219
arti¬cial di¬usion, 373 method of lines
asymptotically optimal, 199 horizontal, 294
BICGSTAB, 238 vertical, 293
block-Gauss“Seidel, 211 method of simultaneous
block-Jacobi, 211 displacements, 203
CG, 221 method of successive displacements,
classical Ritz“Galerkin, 67 204
collocation, 68 MIC decomposition, 232
consistent, 28 micro scale, 5
convergence, 27 minimum angle condition, 141
Crank-Nicolson, 313 minimum principle, 36
Cuthill“McKee, 89 mobility, 10
reverse, 90 molecular di¬usivity, 11
Euler explicit, 313 monotonicity
Euler implicit, 313 inverse, 41, 280
extrapolation, 215 monotonicity test, 357
Index 421

moving front method, 179 equivalent, 395
multi-index, 53, 394 numbering
length, 53, 394 columnwise, 25
order, 53, 394 rowwise, 25
multiblock method, 180
multigrid iteration, 243 octree technique, 177
algorithm, 243 one-step method, 316
multigrid method, 243 A-stable, 317
algebraic, 240 strongly, 319
L-stable, 319
neighbour, 38 nonexpansive, 316
nested iteration, 200, 252 stable, 320
algorithm, 253 one-step-theta method, 312
Neumann™s lemma, 398 operator, 403
Newton™s method, 349 operator norm, 403
algorithm, 357 order of consistency, 28
damped, 357 order of convergence, 27
inexact, 355 orthogonal, 401
simpli¬ed, 353 orthogonality of the error, 68
nodal basis, 61, 125 outer unit normal, 14, 97
nodal value, 58 out¬‚ow boundary, 108
node, 57, 115 overlay method, 177
adjacent, 127 overrelaxation, 209
degree, 89 overshooting, 371
neighbour, 63, 89, 211
norm, 400 parabolic boundary, 325
discrete L2 -, 27 parallelogram identity, 400
equivalence of, 401 Parseval™s identity, 292
Euclidean, 395 particle velocity, 7
Frobenius, 396 partition, 256
induced by a scalar product, 400 partition of unity, 407
p -, 395 PCG
matrix, 395 method, 228, 229
maximum, 395 P´clet number
e
maximum , 27 global, 12, 368
maximum column sum, 396 grid, 372
maximum row sum, 396 local, 269
of an operator, 403 permeability, 8
spectral, 397 perturbation lemma, 398
streamline-di¬usion, 378 phase, 5
stronger, 401 immiscible, 7
total, 396 phase average
vector, 395 extrinsic, 6
µ-weighted, 374 intrinsic, 6
normal derivative, 98 k-phase ¬‚ow, 5
normal equations, 234 (k + 1)-phase system, 5
normed space piezometric head, 8
complete, 404 point
norms boundary, 40
422 Index

close to the boundary, 40 relative permeability, 9
far from the boundary, 40 relaxation method, 207
Poisson equation, 8 relaxation parameter, 207
Dirichlet problem, 19 representative elementary volume, 6
polynomial residual, 188, 189, 201, 244
characteristic, 395 inner, 355
matrix, 394 restriction, 248
pore scale, 5 canonical, 247
pore space, 5 Richards equation, 10
porosity, 6 Richardson method, 206
porous medium, 5 optimal relaxation parameter, 208
porous medium equation, 9 Ritz method, 56
preconditioner, 227 Ritz projection, 304
preconditioning, 207, 227 Ritz“Galerkin method
from the left, 227 classical, 67
from the right, 227 root of equation, 342
preprocessor, 176 Rothe™s method, 294
pressure row sum criterion
global, 10 strict, 204, 398
principle of virtual work, 49 weak, 205, 399
projection 2:1-rule, 181
elliptic, 303, 304
prolongation, 246, 247 saturated, 10
canonical, 246 saturated-unsaturated ¬‚ow, 10
pyramidal function, 62 saturation, 7
saturation concentration, 12
quadrature points, 80 scalar product, 400
quadrature rule, 80, 151 energy, 217
accuracy, 152 Euclidean, 401
Gauss“(Legendre), 153 semi-iterative method, 215
integration points, 151 semidiscrete problem, 295
nodal, 152 semidiscretization, 293
trapezoidal rule, 66, 80, 153 seminorm, 400, 406
weights, 151 separation of variables, 285
quadtree technique, 177 set
closed, 393, 402
range, 343 connected, 394
reaction convex, 394
homogeneous, 13 open, 394
inhomogeneous, 11 set of measure zero, 393
surface, 11 shape function, 59
recovery operator, 193 cubic ansatz on simplex, 121
red mblack ordering, 212 d-polynomial ansatz on cube, 123
reduction strategy, 187 linear ansatz on simplex, 120
reference element, 58 quadratic ansatz on simplex, 121
standard simplicial, 117 simplex
re¬nement barycentre, 119
iterative, 231 degenerate, 117
red/green, 181 face, 117
Index 423

regular d-, 117 superposition principle, 16
sliver element, 179 surface coordinate, 119
smoothing system of equations
barycentric, 181 positive real, 233
Laplacian, 181
weighted barycentric, 181 test function, 47
smoothing property, 239, 250 theorem
smoothing step, 178, 242 of Aubin and Nitsche, 145
a posteriori, 243 of Kahan, 212
a priori, 243 of Lax“Milgram, 93
smoothness requirements, 20 of Ostrowski and Reich, 212
Sobolev space, 54, 94 of Poincar´, 71
e
solid matrix, 5 Trace, 96
solute concentration, 11 Thiessen polygon, 262
solution three-term recursion, 234
classical, 21 time level, 312
of an (initial-) boundary value time step, 312
problem, 17 tortuosity factor, 11
variational, 49 trace, 97
weak, 49, 290 transformation
uniqueness, 51 compatible, 134
solvent, 5 isoparametric, 168
SOR method, 210, 213 transformation formula, 137

<< . .

. 58
( : 59)



. . >>