Polyteknisk Boghandel

This book deals with discretization techniques for elliptic, parabolic and hyperbolic partial differential equations. It provides an introduction to the main principles of discretizations and presents to the reader the ideas and analysis of advanced numerical methods in this area. It is the authors' aim to give mathematically-inclined students, scientists and engineers a textbook that contains all the basic discretization techniques for the three fundamental types of partial differential equations and in which the reader can find analytical tools, properties of discretizations, and some advice on algorithmic aspects. The book also covers recent research developments: for instance, introductions are given to a posteriori error estimation, discontinuous Galerkin methods, and optimal control for partial differential equations---these topics of current interest are rarely considered in other textbooks. While finite element methods are the main focus of the book, finite difference methods and finite volume techniques are also presented. Furthermore, the book provides the basic tools needed to solve the discrete problems generated, while chapters on singularly perturbed problems, variational inequalities and optimal control illuminate special topics that reflect the research interests of the authors.

Contents Notation1 Basics
1.
1 Classification and Correctness1.
2 Fourier's Method, Integral Transforms1.
3 Maximum Principle, Fundamental Solution2 Finite Difference Methods
2.
1 Basic Concepts2.
2 Illustrative Examples
2.
3 Transportation Problems and Conservation Laws
2.
4 Elliptic Boundary Value Problems
2.
5 Finite Volume Methods as Finite Difference Schemes2.
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 NonlinearProblems4 The Finite Element Method
4.
1 A First Example
4.
2 Finite-Element-Spaces4.
3 Practical Aspects of the Finite Element Method
4.
4 Convergence of Conforming Methods4.
5 NonconformingFiniteElementMethods4.
6 Mixed Finite Elements
4.
7 Error Estimators and adaptive FEM
4.
8 The Discontinuous Galerkin Method4.
9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems
5.
1 Parabolic Problems
5.
2 Second-Order Hyperbolic Problems6 Singularly Perturbed Boundary Value Problems
6.
1 Two-Point Boundary Value Problems
6.
2 Parabolic Problems, One-dimensional in Space6.
3 Convection-Diffusion Problems in Several Dimensions7 Variational Inequalities, Optimal Control
7.
1 Analytic Properties7.
2 Discretization of Variational Inequalities7.
3 Penalty Methods7.
4 Optimal Control of PDEs8 Numerical Methods for Discretized Problems8.
1 Some Particular Properties of the Problems8.
2 Direct Methods8.
3 Classical Iterative Methods
8.
4 The Conjugate Gradient Method
8.
5 Multigrid Methods
8.
6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs Bibliography: Original Papers
Index