有限元 原书第3版 英文版PDF电子书下载

Chapter Ⅰ Introduction 1

1．Examples and Classification of PDE's 2

Examples 2

Classification of PDE's 8

Well-posed problems 9

Problems 10

2．The Maximum Principle 12

Examples 13

Corollaries 14

Problem 15

3．Finite Difference Methods 16

Discretization 16

Discrete maximum principle 19

Problem 21

4．A Convergence Theory for Difference Methods 22

Consistency 22

Local and global error 22

Limits of the con-vergence theory 24

Problems 26

Chapter Ⅱ Conforming Finite Elements 27

1．Sobolev Spaces 28

Introduction to Sobolev spaces 29

Friedrichs' inequality 30

Possible singularities of H1 functions 31

Compact imbeddings 32

Problems 33

2．Variational Formulation of Elliptic Boundary-Value Problems of Second Order 34

Variational formulation 35

Reduction to homogeneous bound-ary conditions 36

Existence of solutions 38

Inhomogeneous boundary conditions 42

Problems 42

3．The Neumann Boundary-Value Problem.A Trace Theorem 44

Ellipticity in H1 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 RitzGalerkin Method and Some Finite Elements 53

Model problem 56

Problems 58

5．Some Standard Finite Elements 60

Requirements on the meshes 61

Significance of the differentia-bility properties 62

Triangular elements with complete polyno-mials 64

Remarks on C1 elements 67

Bilinear elements 68

Quadratic rectangular elements 69

Affine families 70

Choice of an element 74

Problems 74

6．Approximation Properties 76

The BrambleHilbert lemma 77

Triangular elements with com-plete polynomials 78

Bilinear quadrilateral elements 81

In-verse estimates 83

Clément's interpolation 84

Appendix:On the optimality of the estimates 85

Problems 87

7．Error Bounds for Elliptic Problems of Second Order 89

Remarks on regularity 89

Error bounds in the energy norm 90

L2 estimates 91

A simple L∞ estimate 93

The L2-projector 94

Problems 95

8．Computational Considerations 97

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 105

1．Abstract Lenmas and a Simple Boundary Approximation 106

Generalizations of Céa'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 117

Isoparametric triangular elements 117

Isoparametric quadrilateral elements 119

Problems 121

3．Further Tools from Functional Analysis 122

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 129

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 145

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 157

Variational formulation 158

The inf-sup condition 159

Nearly incompressible flows 161

Problems 161

7．Finite Elements for the Stokes Problem 162

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 172

Residual estimators 174

Lower estimates 176

Remark on other estimators 179

Local mesh refinement and convergence 179

9．A Posteriori Error Estimates via the Hypercircle Method 181

Chapter Ⅳ The Conjugate Gradient Method 186

1．Classical Iterative Methods for Solving Linear Systems 187

Stationary linear processes 187

The Jacobi and Gauss-Seidel methods 189

The model problem 192

Overrelaxation 193

Problems 195

2．Gradient Methods 196

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 201

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 210

Preconditioning by SSOR 213

Preconditioning by ILU 214

Remarks on parallelization 216

Nonlinear problems 217

Prob-lems 218

5．Saddle Point Problems 221

The Uzawa algorithm and its variants 221

An alternative 223

Problems 224

Chapter Ⅴ Multigrid Methods 225

1．Multigrid Methods for Variational Problems 226

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 237

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 248

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 255

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 261

Schwarz alternating method 262

Assumptions 265

Direct con-sequences 266

Convergence of multiplicative methods 267

Verification of A1 269

Local mesh refinements 270

Problems 271

6．Nonlinear Problems 272

The multigrid-Newton method 273

The nonlinear multigrid method 274

Starting values 276

Problems 277

Chapter Ⅵ Finite Elements in Solid Mechanics 278

1．Introduction to Elasticity Theory 279

Kinematics 279

The equilibrium equations 281

The Piola trans-form 283

Constitutive Equations 284

Linear material laws 288

2．Hyperelastic Materials 290

3．Linear Elasticity Theory 293

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 315

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 323

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 335

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 348

Index 361