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Preface to the Third English Edition page | |

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Preface to the First English Edition | |

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Preface to the German Edition | |

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Notation | |

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Introduction | |

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Examples and Classification of PDE's | |

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Examples | |

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Classification of PDE's | |

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Well-posed Problems | |

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Problems | |

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The Maximum Principle | |

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Examples | |

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Corollaries | |

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Problem | |

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Finite Difference Methods | |

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Discretization | |

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Discrete maximum principle | |

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Problem | |

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A Convergence Theory for Difference Methods | |

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Consistency | |

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Local and global error | |

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Limits of the convergence theory | |

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Problems | |

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Conforming Finite Elements | |

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Sobolev Spaces | |

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Introduction to Sobolev spaces | |

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Friedrichs' inequality | |

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Possible singularities of H1 functions | |

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Compact imbeddings | |

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Problems | |

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Variational Formulation of Elliptic Boundary-Value Problems of Second Order | |

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Variational formulation | |

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Reduction to homogeneous boundary conditions | |

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Existence of solutions | |

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Inhomogeneous boundary conditions | |

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Problems | |

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The Neumann Boundary-Value Problem. A Trace Theorem | |

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Ellipticity in H1 | |

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Boundary-value problems with natural boundary conditions | |

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Neumann boundary conditions | |

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Mixed boundary conditions | |

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Proof of the trace theorem | |

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Practical consequences of the trace theorem | |

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Problems | |

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The Ritz-Galerkin Method and Some Finite Elements | |

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Model Problem | |

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Problems | |

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Some Standard Finite Elements | |

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Requirements on the meshes | |

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Significance of the differentiability properties | |

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Triangular elements with complete polynomials | |

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Remarks on C1 elements | |

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Bilinear elements | |

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Quadratic rectangular elements | |

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Affine families | |

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Choice of an element | |

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Problems | |

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Approximation Properties | |

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The Bramble-Hilbert lemma | |

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Triangular elements with complete polynomials | |

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Bilinear quadrilateral elements | |

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Inverse estimates | |

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Clï¿½ment's interpolation | |

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Appendix: On the optimality of the estimates | |

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Problems | |

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Error Bounds for Elliptic Problems of Second Order | |

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Remarks on regularity | |

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Error bounds in the energy norm | |

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L2 estimates | |

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A simple L∞ estimate | |

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The L2-projector | |

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Problems | |

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Computational Considerations | |

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Assembling the stiffness matrix | |

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Static condensation | |

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Complexity of setting up the matrix | |

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Effect on the choice of a grid | |

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Local mesh refinement | |

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Implementation of the Neumann boundary-value Problem | |

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Problems | |

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Nonconforming and Other Methods | |

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Abstract Lemmas and a Simple Boundary Approximation | |

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Generalizations of Cï¿½a's lemma | |

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Duality methods | |

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The Crouzeix-Raviart element | |

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A Simple approximation to curved boundaries | |

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Modifications of the duality argument | |

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Problems | |

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Isoparametric Elements | |

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Isoparametric triangular elements | |

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Isoparametric quadrilateral elements | |

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Problems | |

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Further Tools from Functional Analysis | |

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Negative norms | |

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Adjoint operators | |

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An abstract existence theorem | |

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An abstract convergence theorem | |

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Proof of Theorem 3.4 | |

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Problems | |

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Saddle Point Problems | |

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Saddle points and minima | |

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The inf-sup condition | |

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Mixed finite element methods | |

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Fortin interpolation | |

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Saddle point problems with penalty term | |

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Typical applications | |

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Problems | |

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Mixed Methods for the Poisson Equation | |

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The Poisson equation as a mixed problem | |

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The Raviart - Thomas element | |

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Interpolation by Raviart-Thomas elements | |

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Implementation and postprocessing | |

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Mesh-dependent norms for the Raviart-Thomas element | |

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The Softening behaviour of mixed methods | |

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Problems | |

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The Stokes Equation | |

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Variational formulation | |

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The inf-sup condition | |

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Nearly incompressible flows | |

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Problems | |

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Finite Elements for the Stokes Problems | |

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An instable element | |

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The Taylor-Hood element | |

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The MINI element | |

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The divergence-free nonconforming P1 element | |

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Problems | |

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A Posteriori Error Estimates | |

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Residual estimators | |

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Lower estimates | |

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Remark on other estimators | |

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Local mesh refinement and convergence | |

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A Posteriori Error Estimates via the Hypercircle Method | |

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The Conjugate Gradient Method | |

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Classical Iterative Methods for Solving Linear Systems | |

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Stationary linear processes | |

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The Jacobi and Gauss-Seidel methods | |

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The model problem | |

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Overrelaxation | |

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Problems | |

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Gradient Methods | |

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The general gradient method | |

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Gradient methods and quadratic functions | |

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Convergence behavior in the case of large condition numbers | |

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Problems | |

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Conjugate Gradient and the Minimal Residual Method | |

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The CG algorithm | |

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Analysis of the CG method as an optimal method | |

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The minimal residual method | |

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Indefinite and unsymmetric matrices | |

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Problems | |

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Preconditioning | |

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Preconditioning by SSOR | |

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Preconditioning by ILU | |

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Remarks on parallelization | |

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Nonlinear Problems | |

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Problems | |

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Saddle Point Problems | |

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The Uzawa algorithm and its variants | |

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An alternative | |

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Problems | |

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Multigrid Methods | |

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Multigrid Methods for Variational Problems | |

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Smoothing properties of classical iterative methods | |

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The multigrid idea | |

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The algorithm | |

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Transfer between grids | |

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Problems | |

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Convergence of Multigrid Methods | |

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Discrete norms | |

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Connection with the Sobolev norm | |

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Approximation property | |

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Convergence proof for the two-grid method | |

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An alternative short proof | |

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Some variants | |

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Problems | |

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Convergence for Several Levels | |

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A recurrence formula for the W-cycle | |

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An improvement for the energy norm | |

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The convergence proof for the V-cycle | |

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Problems | |

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Nested Iteration | |

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Computation of starting values | |

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Complexity | |

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Multigrid methods with a small number of levels | |

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The CASCADE algorithm | |

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Problems | |

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Multigrid Analysis via Space Decomposition | |

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Schwarz alternating method | |

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Assumptions | |

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Direct consequences | |

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Convergence of multiplicative methods | |

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Verification of A1 | |

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Local mesh refinements | |

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Problems | |

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Nonlinear Problems | |

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The multigrid-Newton method | |

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The nonlinear multigrid method | |

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Starting values | |

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Problems | |

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Finite Elements in Solid Mechanics | |

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Introduction to Elasticity Theory | |

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Kinematics | |

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The equilibrium equations | |

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The Piola transform | |

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Constitutive Equations | |

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Linear material laws | |

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Hyperelastic Materials | |

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Linear Elasticity Theory | |

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The variational problem | |

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The displacement formulation | |

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The mixed method of Hellinger and Reissner | |

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The mixed method of Hu and Washizu | |

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Nearly incompressible material | |

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Locking | |

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Locking of the Timoshenko beam and typical remedies | |

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Problems | |

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Membranes | |

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Plane stress states | |

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Plane strain states | |

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Membrane elements | |

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The PEERS element | |

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Problems | |

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Beams and Plates: The Kirchhoff Plate | |

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The hypotheses | |

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Note on beam models | |

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Mixed methods for the Kirchoff plate | |

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DKT elements | |

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Problems | |

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The Mindlin-Reissner Plate | |

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The Helmholtz decomposition | |

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The mixed formulation with the Helmholtz decomposition | |

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MITC elements | |

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The Model without a Helmboltz decomposition | |

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Problems | |

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References | |

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Index | |