有限元(第3版)(英文版)

.boundary conditions 42—problems 42
§ 3. the neumann boundary-value problem. a trace theorem
ellipticity in h 1 44— boundary-value problems with natural bound-ary conditions 45—neumann boundary conditions 46—mixed boundary conditions 47—proof of the trace theorem 48—practi-cal consequences of the trace theorem 50—problems 52
§ 4: the ritz-galerkin method and some finite elements model problem 56—problems 58
§ 5. some standard finite elements
requirements on the meshes 61—significance of the differentia-bility properties 62—triangular elements with complete polyno-mials 64—remarks on cl elements 67—bilinear elements 68 —quadratic rectangular elements 69—affine families 70—choice of an element 74—problems 74
§ 6. approximation properties
the bramble-hilbert lemma 77—triangular elements with com-plete polynomials 78—bilinear quadrilateral elements 81—in-verse estimates 83—c16ment's interpolation 84—appendix: on the optimality of the estimates 85—problems 87
§ 7. error bounds for elliptic problems of second order
remarks on regularity 89—error bounds in the energy norm 90 —l2 estimates 91—a simple loo estimate 93—the l2-projector 94—problems 95
§ 8. computational considerations
assembling the stiffness matrix 97—static condensation 99 —complexity of setting up the matrix 100—effect on the choice of a grid 100—local mesh refinement 100—implementation of the neumann boundary-value problem 102—problems 103
chapter iii nonconforming and other methods
§ 1. abstract lemmas and a simple boundary approximation generalizations of cta's lemma 106- duality methods 108- the crouzeix-raviart element 109 —a simple approximation to curved boundaries 112—modifications of the duality argument 114 —problems 116
§ 2. isoparametric elements
isoparametric triangular elements 117—isoparametric quadrilateral elements 119—problems 121
§ 3. further tools from functional analysis
negative norms 122 —adjoint operators 124 —an abstract exis-tence theorem 124— an abstract convergence theorem 126—proof of theorem 3.4 127- problems 128
§ 4. saddle point problems
saddle points and minima 129—the inf-sup condition 130—mixed finite element methods 134—fortin interpolation 136 —saddle point problems with penalty term 138 —typical applications 141—problems 142
§ 5. mixed methods for the poisson equation
the poisson equation as a mixed problem 145—the raviart- thomas element 148—interpolation by raviart-thomas elements 149—implementation and postprocessing 152—mesh-dependent norms for the raviart-thomas element 153—the softening be-haviour of mixed methods 154—problems 156
§ 6. the stokes equation
variational formulation 158 —the inf-sup condition 159 —nearly incompressible flows 161—problems 161
§ 7. finite elements for the stokes problem
an instable element 162—the taylor-hood element 167—the mini element 168—the divergence-free nonconforming p1 ele-ment 170—problems 171
§ 8. a posteriori error estimates
residual estimators 174— lower estimates 176 m remark on other estimators 179—local mesh refinement and convergence 179
§ 9. a posteriofi error estimates via the hypercircle method
chapter iv the conjugate gradient method
§ 1. classical iterative methods for solving linear systems stationary linear processes 187—the jacobi and gauss-seidel methods 189—the model problem 192—overrelaxation 193 —problems 195
§ 2. gradient methods
the general gradient method 196—gradient methods and quadratic functions 197—convergence behavior in the case of large condition numbers 199—problems 200
§ 3. conjugate gradient and the minimal residual method
the cg algorithm 203—analysis of the cg method as an optimal method 196—the minimal residual method 207—indefinite and unsymmetric matrices 208—problems 209
§ 4. preconditioning
preconditioning by ssor 213—preconditioning by ilu 214 —remarks on parallelization 216—nonlinear problems 217—prob-lems 218
§ 5. saddle point problems
the uzawa algorithm and its variants 221—an alternative 223 —problems 224
chapter v multigrid methods
§ 1. multigrid methods for variational problems
smoothing properties of classical iterative methods 226 —the multi-grid idea 227—the algorithm 228—transfer between grids 232— problems 235
§ 2. convergence of multigrid methods
discrete norms 238—connection with the sobolev norm 240 —approximation property 242—convergence proof for the two-grid method 244—an alternative short proof 245—some variants 245— problems 246
§ 3. convergence for several levels
a recurrence formula for the w-cycle 248—an improvement for the energy norm 249—the convergence proof for the v-cycle 251— problems 254
§ 4. nested iteration
computation of starting values 255—complexity 257—multi-grid methods with a small number of levels 258—the cascade algorithm 259—problems 260
§ 5. multigrid analysis via space decomposition
schwarz alternating method 262—assumptions 265—direct con-sequences 266—convergence of multiplicative methods 267verification of a 1 269—local mesh refinements 270—problems 271
§ 6. nonlinear problems
the multigrid-newton method 273—the nonlinear multigrid method 274—starting values 276—problems 277
chapter vi finite elements in solid mechanics
§ 1. introduction to elasticity theory
kinematics 279—the equilibrium equations 281—the piola trans-form 283—constitutive equations 284—linear material laws 288
§ 2. hyperelastic materials
§ 3. linear elasticity theory
the variational problem 293—the displacement formulation 297
the mixed method of hellinger and reissner 300—the mixed method of hu and washizu 302—nearly incompressible material 304—locking 308—locking of the timoshenko beam and typical remedies 310—problems 314
§ 4. membranes
plane stress states 315—plane strain states 316—membrane ele-ments 316 —the peers element 317—problems 320
§ 5. beams and plates: the kirchhoff plate
the hypotheses 323—note on beam models 326—mixed methods for the kirchoff plate 326—dkt elements 328—problems 334
§ 6. the mindlin-reissner plate
the helmholtz decomposition 336—the mixed formulation with the helmholtz decomposition 338—mitc elements 339—the model without a helmholtz decomposition 343—problems 346 references
index