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Index
Coo(O), CO'(O), 25 Ck,I«O), 26 HS(r),41 HS(fo), HS(fo), 43 HS(O), HS(O), Ho(O), 39 H;w(f),43 H;(O), 71 H;1/2(r), 132
Hy2(r), 137 HH2(f), 140 Lioc(O), 28 Lp(O), Loo(O), 27 L2,o(0), 83 W;(0),30 V(O), V'(O), 35 S(IRd), S'(IR~), 36
A-orthogonale Vektoren, 271 Ableitung
distributionelle, 35 verallgemeinerte, 29
Airysche Spannungsfunktion, 19, 97 Ansatzraum
konformer, 174 konstanter, 216, 245 linearer, 200, 221, 257 quadratischer, 207
Approximation der Linearform, 177 des Operators, 178
Approximationseigenschaft, 176,205, 208, 220, 224
Aubin-Nitsche-Trick, 227, 248, 258
BBL Bedingung, 183 benachbarte Elemente, 189, 214
Bessel-Potential, 38 Bettische Formel
Erste, 16 Zweite, 18
Bilinearform Elastostatik, 16 Potentialgleichung, 13 Stokes-System, 23
Calder6n-Projektor, 130 Cauchy-Daten, 90 CG Verfahren, 275
nach Bramble/Pasciak, 305 vorkonditioniertes, 278
Clement-Operator, 210 Cluster
zulassiges, 311 Clusterbaum, 310
D arstellungsformel Elastostatik, 96 Laplace, 89 Stokes-System, 101, 105
Differentialgleichung Bi-Laplace, 18 Laplace-Operator, 13, 106 Lineare Elastostatik 15 149 , , Stokes-System, 22, 158
Differentialoperator, 11 direkte Methode, 161 Dirichlet-Neumann-Abbildung, 142,
171 Distribution, 35 Doppelschichtpotential, 118, 155
adjungiertes, 114, 151 partielle Integration, 157
Index
Dualitatsprodukt, 27, 42 Dualraum,31 Durchmesser, 188, 214
ebener Spannungszustand, 19 ebener Verschiebungszustand, 20 Einfachschichtpotential, 112, 150 Elastizitatsmodul, 15 elliptisch, 12, 50
Fernfeld, 311 finites Element, 187 Formfunktion
Bubble-, 199 konstante, 195 lineare, 196 quadratische, 198
formregular, 189, 215 Fourier-Transformation, 36 Fundamentallosung, 90
Laplace-Operator, 96 Lineare Elastostatik, 99 Stokes-System, 102
Galerkin-Bubnov-Formulierung, 175 Galerkin-Orthogonalitat, 175, 180, 185 Galerkin-Petrov-Formulierung, 180 Gebietszerlegung, 346 gleichma6ig elliptisch, 12 global gleichma6ig, 189 Gram-Schmidt-Orthogonalisierung, 273 Greensche Formel
Erste, 13 Zweite, 13
Haar-Wavelets,333 hierarchische Matrizen, 312 Hookesches Gesetz, 15 hypersingularer Operator, 121, 156
partielle Integration, 124, 127, 157
Indexmenge, 187 indirekte Methode, 162 inkompressibel, 21
Interpolation, 202 inverse Ungleichung
globale, 202, 222, 225 lokale, 197, 222
Kanten,187 Kapazitat, 135
logarithmische, 135 Kern, 53 Knoten, 187,213 Konditionszahl, 276
361
BEM Steifigkeitsmatrix, 251,260 FEM Steifigkeitsmatrix, 232
konjugierte Vektoren, 271 konjugiertes Gradientenverfahren, 275 Konormalenableitung
Elastostatik, 16 Potentialgleichung, 13 Stokes-System, 24
Kontraktion, 142 Kornsche Ungleichung
Erste,78 Zweite,79
Kriterium von Fortin, 181, 186
Losbarkeitsbedingung, 14, 18, 22, 53 Losungsmannigfaltigkeit, 55 Lagrange
-Funktional, 58 -Multiplikator, 57, 68, 238
Lame-Konstanten, 16 Lastvektor, 232, 250 Lemma
von Bramble-Hilbert, 33, 45 von Cea, 176 von Lax-Milgram, 50 von Schur, 290 von Strang, 178, 179
Lipschitz-Gebiet, 26 lokal gleichma6ig, 189
Maschenweite globale, 189, 213 lokale, 188, 213
362
maximal zulassig, 311 Momentenbedingung, 333 Multiindex, 25 Multilevel-Operator, 288
N ahfeld, 311 natiirliche Dichte, 135, 137 Neumannsche Reihe, 163, 168 Newton-Potential, 106 Normaquivalenz, 195
Oberfiachenrotation, 126 Operatorgleichung, 46 orthogonales Komplement, 53 Orthogonalraum, 53
Projektion Hl_, 204 HU-, 220, 223 L2-, 203, 216, 223
Quasi-Interpolationsoperator, 210 Querkontraktionszahl, 15
Radius, 189 Randbedingungen
Dirichlet-', 14 Gleit-,24 Neumann-, 14 Robin-,14
Randelement, 212 Randspannungsoperator, 17 Randwertproblem
Dirichlet, 65, 79, 162, 226, 245, 307, 346
gemischtes, 73, 83, 169, 263 im AuBenraum, 172 Neumann, 70, 81, 165, 235, 256 Robin, 75, 171, 269
Referenzelement, 190, 193, 214 Reihenentwicklung
ACA, 324 Fundamentallosung, 320 Taylor-Reihe, 317
Riesz-Abbildung, 50 Rotation, 21
Sattelpunktproblem, 57 Satz
Index
closed range theorem, 53 Darstellungssatz von Riesz, 48 Einbettungssatz, 31 Interpolationssatz, 40 inverser Spursatz, 44 Normierungssatz, 31 Spursatz, 44 von GauB-Ostrogradski, 12
Schur-Komplement-System, 299,351 Skalierungsbedingung, 71, 83 Sobolev-Raum, 30 Sobolev-Slobodeckii-Norm, 30, 42 Somigliana-Identitat, 149 Spannungsfunktion von Airy, 19, 97 Spannungstensor, 15 Sprungbedingung
adj. Doppelschichtpotential, 117 Doppelschichtpotential, 120 Einfachschichtpotential, 114
Spur, 12 Stabilitatsbedingung, 55, 60, 68, 164
diskrete, 180, 183, 240, 281, 285 Stokes-Problem, 85
Starrkorperbewegungen, 17, 21 Steifigkeitsmatrix, 231, 250, 259
lokale, 234 Steklov-Poincare Operator, 141, 265 symmetrische Approximation, 266 symmetrische Formulierung, 170, 263
'frager einer Funktion, 25
Ungleichung Erste Kornsche, 78 von Cauchy-Schwarz, 28, 291 von Holder, 27 von Minkowski, 27 von Poincare, 33 Zweite Kornsche, 79
Index
Unterteilung, 187, 212 formregulare, 189 global gleichmaf3ige, 189, 214 lokal gleichmaf3ige, 189, 214 zulassige, 188, 213
Variationsformulierung, 46 Verzerrungstensor, 15
Volumen, 188, 213 Vorkonditionierung, 277, 279
mit Integraloperatoren, 282 Multilevel-, 286
Zerlegung der Eins, 236 zulassig
maximal, 311
363