《有限元(第3版)》由布拉艾斯所著，This definitive introduction to finite element methods has been thoroughly updated for this third edition， which features important new material for both research and application of the finite element method.

The discussion of saddle point problems is a lughlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena.……

Preface to the Third English Edition

Preface to the First English Edition

Preface to the German Edition

Notation

                  Chapter I

                 Introduction

1. Examples and Classification of PDE's

Examples 2 -- Classification of PDE's 8 -- Well-posed problems 9

-- Problems 10

2. The Maximum Principle

Examples 13- Corollaries 14- Problem 15

3. Finite Difference Methods

Discretization 16 -- Discrete maximum principle 19 -- Problem 21

4. A Convergence Theory for Difference Methods

Consistency 22 -- Local and global error 22 -- Limits of the con-

vergence theory 24 -- Problems 26

                 Chapter H

             Conforming Finite Elements

1. Sobolev Spaces

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

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

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 C1 elements 67 -- Bilinear elements 68 --

Quadratic rectangular elements 69 -- Affine families 70 -- Choice

of an element 74 -- Problems 74

6. Approximation Properties

The Bramble-Hiibert lemma 77 -- Triangular elements with com-

plete polynomials 78 -- Bilinear quadrilateral elements 81 -- In-

verse estimates 83 -- CIement'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 L∞ 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 C6a'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 -- Remark on other

estimators 179 -- Local mesh refinement and convergence 179

9. A Posteriori 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 267 --

 Verification of A1 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