有限元 原书第3版 英文版PDF|Epub|txt|kindle电子书版本网盘下载

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Chapter Ⅰ Introduction1

1．Examples and Classification of PDE's2

Examples2

Classification of PDE's8

Well-posed problems9

Problems10

2．The Maximum Principle12

Examples13

Corollaries14

Problem15

3．Finite Difference Methods16

Discretization16

Discrete maximum principle19

Problem21

4．A Convergence Theory for Difference Methods22

Consistency22

Local and global error22

Limits of the con-vergence theory24

Problems26

Chapter Ⅱ Conforming Finite Elements27

1．Sobolev Spaces28

Introduction to Sobolev spaces29

Friedrichs' inequality30

Possible singularities of H1 functions31

Compact imbeddings32

Problems33

2．Variational Formulation of Elliptic Boundary-Value Problems of Second Order34

Variational formulation35

Reduction to homogeneous bound-ary conditions36

Existence of solutions38

Inhomogeneous boundary conditions42

Problems42

3．The Neumann Boundary-Value Problem.A Trace Theorem44

Ellipticity in H144

Boundary-value problems with natural bound-ary conditions45

Neumann boundary conditions46

Mixed boundary conditions47

Proof of the trace theorem48

Practi-cal consequences of the trace theorem50

Problems52

4．The RitzGalerkin Method and Some Finite Elements53

Model problem56

Problems58

5．Some Standard Finite Elements60

Requirements on the meshes61

Significance of the differentia-bility properties62

Triangular elements with complete polyno-mials64

Remarks on C1 elements67

Bilinear elements68

Quadratic rectangular elements69

Affine families70

Choice of an element74

Problems74

6．Approximation Properties76

The BrambleHilbert lemma77

Triangular elements with com-plete polynomials78

Bilinear quadrilateral elements81

In-verse estimates83

Clément's interpolation84

Appendix:On the optimality of the estimates85

Problems87

7．Error Bounds for Elliptic Problems of Second Order89

Remarks on regularity89

Error bounds in the energy norm90

L2 estimates91

A simple L∞ estimate93

The L2-projector94

Problems95

8．Computational Considerations97

Assembling the stiffness matrix97

Static condensation99

Complexity of setting up the matrix100

Effect on the choice of a grid 100 Local mesh refinement100

Implementation of the Neumann boundary-value problem102

Problems103

Chapter Ⅲ Nonconforming and Other Methods105

1．Abstract Lenmas and a Simple Boundary Approximation106

Generalizations of Céa's lemma106

Duality methods108

The Crouzeix-Raviart element109

A simple approximation to curved boundaries112

Modifications of the duality argument114

Problems116

2．Isoparametric Elements117

Isoparametric triangular elements117

Isoparametric quadrilateral elements119

Problems121

3．Further Tools from Functional Analysis122

Negative norms122

Adjoint operators124

An abstract exis-tence theorem124

An abstract convergence theorem126

Proof of Theorem 3.4127

Problems128

4．Saddle Point Problems129

Saddle points and minima129

The inf-sup condition130

Mixed finite element methods134

Fortin interpolation136

Saddle point problems with penalty term138

Typical applications141

Problems142

5．Mixed Methods for the Poisson Equation145

The Poisson equation as a mixed problem145

The Raviart-Thomas element148

Interpolation by Raviart-Thomas elements149

Implementation and postprocessing152

Mesh-dependent norms for the Raviart-Thomas element153

The softening be-haviour of mixed methods154

Problems156

6．The Stokes Equation157

Variational formulation158

The inf-sup condition159

Nearly incompressible flows161

Problems161

7．Finite Elements for the Stokes Problem162

An instable element162

The Taylor-Hood element167

The MINI element168

The divergence-free nonconforming P1 ele-ment170

Problems171

8．A Posteriori Error Estimates172

Residual estimators174

Lower estimates176

Remark on other estimators179

Local mesh refinement and convergence179

9．A Posteriori Error Estimates via the Hypercircle Method181

Chapter Ⅳ The Conjugate Gradient Method186

1．Classical Iterative Methods for Solving Linear Systems187

Stationary linear processes187

The Jacobi and Gauss-Seidel methods189

The model problem192

Overrelaxation193

Problems195

2．Gradient Methods196

The general gradient method196

Gradient methods and quadratic functions197

Convergence behavior in the case of large condition numbers199

Problems200

3．Conjugate Gradient and the Minimal Residual Method201

The CG algorithm203

Analysis of the CG method as an optimal method196

The minimal residual method207

Indefinite and unsymmetric matrices208

Problems209

4．Preconditioning210

Preconditioning by SSOR213

Preconditioning by ILU214

Remarks on parallelization216

Nonlinear problems217

Prob-lems218

5．Saddle Point Problems221

The Uzawa algorithm and its variants221

An alternative223

Problems224

Chapter Ⅴ Multigrid Methods225

1．Multigrid Methods for Variational Problems226

Smoothing properties of classical iterative methods226

The multi-grid idea227

The algorithm228

Transfer between grids232

Problems235

2．Convergence of Multigrid Methods237

Discrete norms238

Connection with the Sobolev norm240

Approximation property242

Convergence proof for the two-grid method244

An alternative short proof245

Some variants245

Problems246

3．Convergence for Several Levels248

A recurrence formula for the W-cycle248

An improvement for the energy norm249

The convergence proof for the V-cycle251

Problems254

4．Nested Iteration255

Computation of starting values255

Complexity257

Multi-grid methods with a small number of levels258

The CASCADE algorithm259

Problems260

5．Multigrid Analysis via Space Decomposition261

Schwarz alternating method262

Assumptions265

Direct con-sequences266

Convergence of multiplicative methods267

Verification of A1269

Local mesh refinements270

Problems271

6．Nonlinear Problems272

The multigrid-Newton method273

The nonlinear multigrid method274

Starting values276

Problems277

Chapter Ⅵ Finite Elements in Solid Mechanics278

1．Introduction to Elasticity Theory279

Kinematics279

The equilibrium equations281

The Piola trans-form283

Constitutive Equations284

Linear material laws288

2．Hyperelastic Materials290

3．Linear Elasticity Theory293

The variational problem293

The displacement formulation297

The mixed method of Hellinger and Reissner300

The mixed method of Hu and Washizu302

Nearly incompressible material304

Locking308

Locking of the Timoshenko beam and typical remedies310

Problems314

4．Membranes315

Plane stress states315

Plane strain states316

Membrane ele-ments316

The PEERS element317

Problems320

5．Beams and Plates:The Kirchhoff Plate323

The hypotheses323

Note on beam models326

Mixed methods for the Kirchoff plate326

DKT elements328

Problems334

6．The Mindlin-Reissner Plate335

The Helmholtz decomposition336

The mixed formulation with the Helmholtz decomposition338

MITC elements339

The model without a Helmholtz decomposition343

Problems346

References348

Index361