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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