Numerical treatment of partial differential equations

Many well-known models in the natural sciences and engineering, and today even in economics, depend on partial di?erential equations. Thus the e?cient numerical solution of such equations plays an ever-increasing role in state-- the-art technology. This demand and the computational power available from current computer hardware have together stimulated the rapid development of numerical methods for partial di?erential equations--a development that encompasses convergence analyses and implementational aspects of software packages. In 1988 we started work on the ?rst German edition of our book, which appeared in 1992. Our aim was to give students a textbook that contained the basic concepts and ideas behind most numerical methods for partial di?er- tial equations. The success of this ?rst edition and the second edition in 1994 encouraged us, ten years later, to write an almost completely new version, taking into account comments from colleagues and students and drawing on the enormous progress made in the numerical analysis of partial di?erential equations in recent times. The present English version slightly improves the third German edition of 2005: we have corrected some minor errors and added additional material and references.

Author Notes

Christian Grossmann, born in 1946, is Professor of Numerical Analysis at the TU Dresden
Hans-Gorg Roos, born in 1949, is Professor of Numerical Mathematics at the TU Dresden

Notation	p. XI
1 Basics	p. 1
1.1 Classification and Correctness	p. 1
1.2 Fourier' s Method, Integral Transforms	p. 5
1.3 Maximum Principle, Fundamental Solution	p. 9
1.3.1 Elliptic Boundary Value Problems	p. 9
1.3.2 Parabolic Equations and Initial-Boundary Value Problems	p. 15
1.3.3 Hyperbolic Initial and Initial-Boundary Value Problems	p. 18
2 Finite Difference Methods	p. 23
2.1 Basic Concepts	p. 23
2.2 Illustrative Examples	p. 31
2.3 Transportation Problems and Conservation Laws	p. 36
2.3.1 The One-Dimensional Linear Case	p. 37
2.3.2 Properties of Nonlinear Conservation Laws	p. 48
2.3.3 Difference Methods for Nonlinear Conservation Laws	p. 53
2.4 Elliptic Boundary Value Problems	p. 61
2.4.1 Elliptic Boundary Value Problems	p. 61
2.4.2 The Classical Approach to Finite Difference Methods	p. 62
2.4.3 Discrete Green's Function	p. 74
2.4.4 Difference Stencils and Discretization in General Domains	p. 76
2.4.5 Mixed Derivatives, Fourth Order Operators	p. 82
2.4.6 Local Grid Refinements	p. 89
2.5 Finite Volume Methods as Finite Difference Schemes	p. 90
2.6 Parabolic Initial-Boundary Value Problems	p. 103
2.6.1 Problems in One Space Dimension	p. 104
2.6.2 Problems in Higher Space Dimensions	p. 109
2.6.3 Semi-Discretization	p. 113
2.7 Second-Order Hyperbolic Problems	p. 118
3 Weak Solutions	p. 125
3.1 Introduction	p. 125
3.2 Adapted Function Spaces	p. 128
3.3 Variational Equations and Conforming Approximation	p. 142
3.4 Weakening V-ellipticity	p. 163
3.5 Nonlinear Problems	p. 167
4 The Finite Element Method	p. 173
4.1 A First Example	p. 173
4.2 Finite-Element-Spaces	p. 178
4.2.1 Local and Global Properties	p. 178
4.2.2 Examples of Finite Element Spaces in R[superscript 2] and R[superscript 3]	p. 189
4.3 Practical Aspects of the Finite Element Method	p. 202
4.3.1 Structure of a Finite Element Code	p. 202
4.3.2 Description of the Problem	p. 203
4.3.3 Generation of the Discrete Problem	p. 205
4.3.4 Mesh Generation and Manipulation	p. 210
4.4 Convergence of Conforming Methods	p. 217
4.4.1 Interpolation and Projection Error in Sobolev Spaces	p. 217
4.4.2 Hilbert Space Error Estimates	p. 227
4.4.3 Inverse Inequalities and Pointwise Error Estimates	p. 232
4.5 Nonconforming Finite Element Methods	p. 238
4.5.1 Introduction	p. 238
4.5.2 Ansatz Spaces with Low Smoothness	p. 239
4.5.3 Numerical Integration	p. 244
4.5.4 The Finite Volume Method Analysed from a Finite Element Viewpoint	p. 251
4.5.5 Remarks on Curved Boundaries	p. 254
4.6 Mixed Finite Elements	p. 258
4.6.1 Mixed Variational Equations and Saddle Points	p. 258
4.6.2 Conforming Approximation of Mixed Variational Equations	p. 265
4.6.3 Weaker Regularity for the Poisson and Biharmonic Equations	p. 272
4.6.4 Penalty Methods and Modified Lagrange Functions	p. 277
4.7 Error Estimators and Adaptive FEM	p. 287
4.7.1 The Residual Error Estimator	p. 288
4.7.2 Averaging and Goal-Oriented Estimators	p. 292
4.8 The Discontinuous Galerkin Method	p. 294
4.8.1 The Primal Formulation for a Reaction-Diffusion Problem	p. 295
4.8.2 First-Order Hyperbolic Problems	p. 299
4.8.3 Error Estimates for a Convection-Diffusion Problem	p. 302
4.9 Further Aspects of the Finite Element Method	p. 306
4.9.1 Conditioning of the Stiffness Matrix	p. 306
4.9.2 Eigenvalue Problems	p. 307
4.9.3 Superconvergence	p. 310
4.9.4 p- and hp-Versions	p. 314
5 Finite Element Methods for Unsteady Problems	p. 317
5.1 Parabolic Problems	p. 317
5.1.1 On the Weak Formulation	p. 317
5.1.2 Semi-Discretization by Finite Elements	p. 321
5.1.3 Temporal Discretization by Standard Methods	p. 330
5.1.4 Temporal Discretization with Discontinuous Galerkin Methods	p. 337
5.1.5 Rothe's Method	p. 343
5.1.6 Error Control	p. 347
5.2 Second-Order Hyperbolic Problems	p. 356
5.2.1 Weak Formulation of the Problem	p. 356
5.2.2 Semi-Discretization by Finite Elements	p. 358
5.2.3 Temporal Discretization	p. 363
5.2.4 Rothe's Method for Hyperbolic Problems	p. 368
5.2.5 Remarks on Error Control	p. 372
6 Singularly Perturbed Boundary Value Problems	p. 375
6.1 Two-Point Boundary Value Problems	p. 376
6.1.1 Analytical Behaviour of the Solution	p. 376
6.1.2 Discretization on Standard Meshes	p. 383
6.1.3 Layer-adapted Meshes	p. 394
6.2 Parabolic Problems, One-dimensional in Space	p. 399
6.2.1 The Analytical Behaviour of the Solution	p. 399
6.2.2 Discretization	p. 401
6.3 Convection-Diffusion Problems in Several Dimensions	p. 406
6.3.1 Analysis of Elliptic Convection-Diffusion Problems	p. 406
6.3.2 Discretization on Standard Meshes	p. 412
6.3.3 Layer-adapted Meshes	p. 427
6.3.4 Parabolic Problems, Higher-Dimensional in Space	p. 430
7 Variational Inequalities, Optimal Control	p. 435
7.1 Analytic Properties	p. 435
7.2 Discretization of Variational Inequalities	p. 447
7.3 Penalty Methods	p. 457
7.3.1 Basic Concept of Penalty Methods	p. 457
7.3.2 Adjustment of Penalty and Discretization Parameters	p. 473
7.4 Optimal Control of PDEs	p. 480
7.4.1 Analysis of an Elliptic Model Problem	p. 480
7.4.2 Discretization by Finite Element Methods	p. 489
8 Numerical Methods for Discretized Problems	p. 499
8.1 Some Particular Properties of the Problems	p. 499
8.2 Direct Methods	p. 502
8.2.1 Gaussian Elimination for Banded Matrices	p. 502
8.2.2 Fast Solution of Discrete Poisson Equations, FFT	p. 504
8.3 Classical Iterative Methods	p. 510
8.3.1 Basic Structure and Convergence	p. 510
8.3.2 Jacobi and Gauss-Seidel Methods	p. 514
8.3.3 Block Iterative Methods	p. 520
8.3.4 Relaxation and Splitting Methods	p. 524
8.4 The Conjugate Gradient Method	p. 530
8.4.1 The Basic Idea, Convergence Properties	p. 530
8.4.2 Preconditioned CG Methods	p. 538
8.5 Multigrid Methods	p. 548
8.6 Domain Decomposition, Parallel Algorithms	p. 560
Bibliography: Textbooks and Monographs	p. 571
Bibliography: Original Papers	p. 577
Index	p. 585

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