Preface to the Third English Edition page |
Preface to the First English Edition |
Preface to the German Edition |
Notation |
Introduction / Chapter I: |
Examples and Classification of PDE's / 1: |
Examples |
Classification of PDE's |
Well-posed Problems |
Problems |
The Maximum Principle / 2: |
Corollaries |
Problem |
Finite Difference Methods / 3: |
Discretization |
Discrete maximum principle |
A Convergence Theory for Difference Methods / 4: |
Consistency |
Local and global error |
Limits of the convergence theory |
Conforming Finite Elements / Chapter II: |
Sobolev Spaces |
Introduction to Sobolev spaces |
Friedrichs' inequality |
Possible singularities of H1 functions |
Compact imbeddings |
Variational Formulation of Elliptic Boundary-Value Problems of Second Order |
Variational formulation |
Reduction to homogeneous boundary conditions |
Existence of solutions |
Inhomogeneous boundary conditions |
The Neumann Boundary-Value Problem. A Trace Theorem |
Ellipticity in H1 |
Boundary-value problems with natural boundary conditions |
Neumann boundary conditions |
Mixed boundary conditions |
Proof of the trace theorem |
Practical consequences of the trace theorem |
The Ritz-Galerkin Method and Some Finite Elements |
Model Problem |
Some Standard Finite Elements / 5: |
Requirements on the meshes |
Significance of the differentiability properties |
Triangular elements with complete polynomials |
Remarks on C1 elements |
Bilinear elements |
Quadratic rectangular elements |
Affine families |
Choice of an element |
Approximation Properties / 6: |
The Bramble-Hilbert lemma |
Bilinear quadrilateral elements |
Inverse estimates |
Clément's interpolation |
Appendix: On the optimality of the estimates |
Error Bounds for Elliptic Problems of Second Order / 7: |
Remarks on regularity |
Error bounds in the energy norm |
L2 estimates |
A simple L? estimate |
The L2-projector |
Computational Considerations / 8: |
Assembling the stiffness matrix |
Static condensation |
Complexity of setting up the matrix |
Effect on the choice of a grid |
Local mesh refinement |
Implementation of the Neumann boundary-value Problem |
Nonconforming and Other Methods / Chapter III: |
Abstract Lemmas and a Simple Boundary Approximation |
Generalizations of Céa's lemma |
Duality methods |
The Crouzeix-Raviart element |
A Simple approximation to curved boundaries |
Modifications of the duality argument |
Isoparametric Elements |
Isoparametric triangular elements |
Isoparametric quadrilateral elements |
Further Tools from Functional Analysis |
Negative norms |
Adjoint operators |
An abstract existence theorem |
An abstract convergence theorem |
Proof of Theorem 3.4 |
Saddle Point Problems |
Saddle points and minima |
The inf-sup condition |
Mixed finite element methods |
Fortin interpolation |
Saddle point problems with penalty term |
Typical applications |
Mixed Methods for the Poisson Equation |
The Poisson equation as a mixed problem |
The Raviart - Thomas element |
Interpolation by Raviart-Thomas elements |
Implementation and postprocessing |
Mesh-dependent norms for the Raviart-Thomas element |
The Softening behaviour of mixed methods |
The Stokes Equation |
Nearly incompressible flows |
Finite Elements for the Stokes Problems |
An instable element |
The Taylor-Hood element |
The MINI element |
The divergence-free nonconforming P1 element |
A Posteriori Error Estimates |
Residual estimators |
Lower estimates |
Remark on other estimators |
Local mesh refinement and convergence |
A Posteriori Error Estimates via the Hypercircle Method / 9: |
The Conjugate Gradient Method / Chapter IV: |
Classical Iterative Methods for Solving Linear Systems |
Stationary linear processes |
The Jacobi and Gauss-Seidel methods |
The model problem |
Overrelaxation |
Gradient Methods |
The general gradient method |
Gradient methods and quadratic functions |
Convergence behavior in the case of large condition numbers |
Conjugate Gradient and the Minimal Residual Method |
The CG algorithm |
Analysis of the CG method as an optimal method |
The minimal residual method |
Indefinite and unsymmetric matrices |
Preconditioning |
Preconditioning by SSOR |
Preconditioning by ILU |
Remarks on parallelization |
Nonlinear Problems |
The Uzawa algorithm and its variants |
An alternative |
Multigrid Methods / Chapter V: |
Multigrid Methods for Variational Problems |
Smoothing properties of classical iterative methods |
The multigrid idea |
The algorithm |
Transfer between grids |
Convergence of Multigrid Methods |
Discrete norms |
Connection with the Sobolev norm |
Approximation property |
Convergence proof for the two-grid method |
An alternative short proof |
Some variants |
Convergence for Several Levels |
A recurrence formula for the W-cycle |
An improvement for the energy norm |
The convergence proof for the V-cycle |
Nested Iteration |
Computation of starting values |
Complexity |
Multigrid methods with a small number of levels |
The CASCADE algorithm |
Multigrid Analysis via Space Decomposition |
Schwarz alternating method |
Assumptions |
Direct consequences |
Convergence of multiplicative methods |
Verification of A1 |
Local mesh refinements |
The multigrid-Newton method |
The nonlinear multigrid method |
Starting values |
Finite Elements in Solid Mechanics / Chapter VI: |
Introduction to Elasticity Theory |
Kinematics |
The equilibrium equations |
The Piola transform |
Constitutive Equations |
Linear material laws |
Hyperelastic Materials |
Linear Elasticity Theory |
The variational problem |
The displacement formulation |
The mixed method of Hellinger and Reissner |
The mixed method of Hu and Washizu |
Nearly incompressible material |
Locking |
Locking of the Timoshenko beam and typical remedies |
Membranes |
Plane stress states |
Plane strain states |
Membrane elements |
The PEERS element |
Beams and Plates: The Kirchhoff Plate |
The hypotheses |
Note on beam models |
Mixed methods for the Kirchoff plate |
DKT elements |
The Mindlin-Reissner Plate |
The Helmholtz decomposition |
The mixed formulation with the Helmholtz decomposition |
MITC elements |
The Model without a Helmboltz decomposition |
References |
Index |
Preface to the Third English Edition page |
Preface to the First English Edition |
Preface to the German Edition |