Numerical treatment of partial differential equations / Christian Grossmann, Hans-Görg Roos ; translated and revised by Martin Stynes.

Format:

Book

Author/Creator:

Grossmann, Christian.

Contributor:

Roos, Hans-Görg, 1949-

Stynes, M. (Martin), 1951-

Series:

Universitext

Standardized Title:

Numerische Behandlung Partieller Differentialgleichungen. English

Language:

English

German

Subjects (All):

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Finite differences.

Finite element method.

Numerical analysis.

Physical Description:

xii, 591 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, 2007.

Summary:

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:

1.1 Classification and Correctness 1

1.2 Fourier' s Method, Integral Transforms 5

1.3 Maximum Principle, Fundamental Solution 9

1.3.1 Elliptic Boundary Value Problems 9

1.3.2 Parabolic Equations and Initial-Boundary Value Problems 15

1.3.3 Hyperbolic Initial and Initial-Boundary Value Problems 18

2 Finite Difference Methods 23

2.3 Transportation Problems and Conservation Laws 36

2.3.1 The One-Dimensional Linear Case 37

2.3.2 Properties of Nonlinear Conservation Laws 48

2.3.3 Difference Methods for Nonlinear Conservation Laws 53

2.4 Elliptic Boundary Value Problems 61

2.4.1 Elliptic Boundary Value Problems 61

2.4.2 The Classical Approach to Finite Difference Methods 62

2.4.3 Discrete Green's Function 74

2.4.4 Difference Stencils and Discretization in General Domains 76

2.4.5 Mixed Derivatives, Fourth Order Operators 82

2.4.6 Local Grid Refinements 89

2.5 Finite Volume Methods as Finite Difference Schemes 90

2.6 Parabolic Initial-Boundary Value Problems 103

2.6.1 Problems in One Space Dimension 104

2.6.2 Problems in Higher Space Dimensions 109

2.6.3 Semi-Discretization 113

2.7 Second-Order Hyperbolic Problems 118

3 Weak Solutions 125

3.2 Adapted Function Spaces 128

3.3 Variational Equations and Conforming Approximation 142

3.4 Weakening V-ellipticity 163

3.5 Nonlinear Problems 167

4 The Finite Element Method 173

4.1 A First Example 173

4.2 Finite-Element-Spaces 178

4.2.1 Local and Global Properties 178

4.2.2 Examples of Finite Element Spaces in R[superscript 2] and R[superscript 3] 189

4.3 Practical Aspects of the Finite Element Method 202

4.3.1 Structure of a Finite Element Code 202

4.3.2 Description of the Problem 203

4.3.3 Generation of the Discrete Problem 205

4.3.4 Mesh Generation and Manipulation 210

4.4 Convergence of Conforming Methods 217

4.4.1 Interpolation and Projection Error in Sobolev Spaces 217

4.4.2 Hilbert Space Error Estimates 227

4.4.3 Inverse Inequalities and Pointwise Error Estimates 232

4.5 Nonconforming Finite Element Methods 238

4.5.2 Ansatz Spaces with Low Smoothness 239

4.5.3 Numerical Integration 244

4.5.4 The Finite Volume Method Analysed from a Finite Element Viewpoint 251

4.5.5 Remarks on Curved Boundaries 254

4.6 Mixed Finite Elements 258

4.6.1 Mixed Variational Equations and Saddle Points 258

4.6.2 Conforming Approximation of Mixed Variational Equations 265

4.6.3 Weaker Regularity for the Poisson and Biharmonic Equations 272

4.6.4 Penalty Methods and Modified Lagrange Functions 277

4.7 Error Estimators and Adaptive FEM 287

4.7.1 The Residual Error Estimator 288

4.7.2 Averaging and Goal-Oriented Estimators 292

4.8 The Discontinuous Galerkin Method 294

4.8.1 The Primal Formulation for a Reaction-Diffusion Problem 295

4.8.2 First-Order Hyperbolic Problems 299

4.8.3 Error Estimates for a Convection-Diffusion Problem 302

4.9 Further Aspects of the Finite Element Method 306

4.9.1 Conditioning of the Stiffness Matrix 306

4.9.2 Eigenvalue Problems 307

4.9.3 Superconvergence 310

4.9.4 p- and hp-Versions 314

5 Finite Element Methods for Unsteady Problems 317

5.1 Parabolic Problems 317

5.1.1 On the Weak Formulation 317

5.1.2 Semi-Discretization by Finite Elements 321

5.1.3 Temporal Discretization by Standard Methods 330

5.1.4 Temporal Discretization with Discontinuous Galerkin Methods 337

5.1.5 Rothe's Method 343

5.1.6 Error Control 347

5.2 Second-Order Hyperbolic Problems 356

5.2.1 Weak Formulation of the Problem 356

5.2.2 Semi-Discretization by Finite Elements 358

5.2.3 Temporal Discretization 363

5.2.4 Rothe's Method for Hyperbolic Problems 368

5.2.5 Remarks on Error Control 372

6 Singularly Perturbed Boundary Value Problems 375

6.1 Two-Point Boundary Value Problems 376

6.1.1 Analytical Behaviour of the Solution 376

6.1.2 Discretization on Standard Meshes 383

6.1.3 Layer-adapted Meshes 394

6.2 Parabolic Problems, One-dimensional in Space 399

6.2.1 The Analytical Behaviour of the Solution 399

6.2.2 Discretization 401

6.3 Convection-Diffusion Problems in Several Dimensions 406

6.3.1 Analysis of Elliptic Convection-Diffusion Problems 406

6.3.2 Discretization on Standard Meshes 412

6.3.3 Layer-adapted Meshes 427

6.3.4 Parabolic Problems, Higher-Dimensional in Space 430

7 Variational Inequalities, Optimal Control 435

7.1 Analytic Properties 435

7.2 Discretization of Variational Inequalities 447

7.3 Penalty Methods 457

7.3.1 Basic Concept of Penalty Methods 457

7.3.2 Adjustment of Penalty and Discretization Parameters 473

7.4 Optimal Control of PDEs 480

7.4.1 Analysis of an Elliptic Model Problem 480

7.4.2 Discretization by Finite Element Methods 489

8 Numerical Methods for Discretized Problems 499

8.1 Some Particular Properties of the Problems 499

8.2 Direct Methods 502

8.2.1 Gaussian Elimination for Banded Matrices 502

8.2.2 Fast Solution of Discrete Poisson Equations, FFT 504

8.3 Classical Iterative Methods 510

8.3.1 Basic Structure and Convergence 510

8.3.2 Jacobi and Gauss-Seidel Methods 514

8.3.3 Block Iterative Methods 520

8.3.4 Relaxation and Splitting Methods 524

8.4 The Conjugate Gradient Method 530

8.4.1 The Basic Idea, Convergence Properties 530

8.4.2 Preconditioned CG Methods 538

8.5 Multigrid Methods 548

8.6 Domain Decomposition, Parallel Algorithms 560

Bibliography: Textbooks and Monographs 571

Bibliography: Original Papers 577.

Notes:

Includes bibliographical references (pages [571]-583) and index.

ISBN:

3540715827

9783540715825

OCLC:

185096377