This book deals with discretization techniques for partial differential equations of elliptic, parabolic and hyperbolic type. It provides an introduction to the main principles of discretization and gives a presentation of the ideas and analysis of advanced numerical methods in the area. The book is mainly dedicated to finite element methods, but it also discusses difference methods and finite volume techniques.

Coverage offers analytical tools, properties of discretization techniques and hints to algorithmic aspects. It also guides readers to current developments in research.

Contents Notation 1 Basics
1.1 Classification and Correctness
1.2 Fourier's Method, Integral Transforms
1.3 Maximum Principle, Fundamental Solution 2 Finite Difference Methods
2.1 Basic Concepts
2.2 Illustrative Examples
2.3 Transportation Problems and Conservation Laws
2.4 Elliptic Boundary Value Problems
2.5 Finite Volume Methods as Finite Difference Schemes
2.6 Parabolic Initial-Boundary Value Problems
2.7 Second-Order Hyperbolic Problems 3 Weak Solutions
3.1 Introduction
3.2 Adapted Function Spaces
3.3 VariationalEquationsand conformingApproximation
3.4 WeakeningV-ellipticity
3.5 NonlinearProblems 4 The Finite Element Method
4.1 A First Example
4.2 Finite-Element-Spaces
4.3 Practical Aspects of the Finite Element Method
4.4 Convergence of Conforming Methods
4.5 NonconformingFiniteElementMethods
4.6 Mixed Finite Elements
4.7 Error Estimators and adaptive FEM
4.8 The Discontinuous Galerkin Method
4.9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems
5.1 Parabolic Problems
5.2 Second-Order Hyperbolic Problems 6 Singularly Perturbed Boundary Value Problems
6.1 Two-Point Boundary Value Problems
6.2 Parabolic Problems, One-dimensional in Space
6.3 Convection-Diffusion Problems in Several Dimensions 7 Variational Inequalities, Optimal Control
7.1 Analytic Properties
7.2 Discretization of Variational Inequalities
7.3 Penalty Methods
7.4 Optimal Control of PDEs 8 Numerical Methods for Discretized Problems
8.1 Some Particular Properties of the Problems
8.2 Direct Methods
8.3 Classical Iterative Methods
8.4 The Conjugate Gradient Method
8.5Multigrid Methods
8.6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs
Bibliography: Original Papers
Index