| 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 |
| Notation |
| Introduction / Chapter I： |
| Examples and Classification of PDE's / 1： |
