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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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| Cl�ment'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 C�a'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 | |