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Numerical Methods for Partial Differential Equations: An Introduction
Vitoriano Ruas, Sorbonne Universités, UPMC - Université Paris 6, France
A comprehensive overview of techniques for the computational solution of PDE's
Numerical Methods for Partial Differential Equations: An Introduction covers the three most popular methods for solving partial differential equations: the finite difference method, the finite element method and the finite volume method. The book combines clear descriptions of the three methods, their reliability, and practical implementation aspects. Justifications for why numerical methods for the main classes of PDE's work or not, or how well they work, are supplied and exemplified.
Aimed primarily at students of Engineering, Mathematics, Computer Science, Physics and Chemistry among others this book offers a substantial insight into the principles numerical methods in this class of problems are based upon. The book can also be used as a reference for research work on numerical methods for PDE's.
Key features:
A balanced emphasis is given to both practical considerations and a rigorous mathematical treatment
The reliability analyses for the three methods are carried out in a unified framework and in a structured and visible manner, for the basic types of PDE's
Special attention is given to low order methods, as practitioner's overwhelming default options for everyday use
New techniques are employed to derive known results, thereby simplifying their proof
Supplementary material is available from a companion website.
Contents
Preface by Eugenio Õnate xi
Preface by Larisa Beilina xiii
Acknowledgements xv
About the Companion Website xvii
Introduction xix
Key Reminders on Linear Algebra xxvii
1 Getting Started in One Space Variable 1
1.1 A Model Two-point Boundary Value Problem 2
1.2 The Basic FDM 7
1.3 The Piecewise Linear FEM (P 1 FEM) 12
1.4 The Basic FVM 17
1.4.1 The Vertex-centred FVM 17
1.4.2 The Cell-centred FVM 20
1.4.3 Connections to the Other Methods 22
1.5 Handling Nonzero Boundary Conditions 24
1.6 Effective Resolution 25
1.6.1 Solving SLAEs for one-dimensional problems 26
1.6.2 Example 1.1: Numerical Experiments with the Cell-centred FVM 27
1.7 Exercises 28
2 Qualitative Reliability Analysis 30
2.1 Norms and Inner Products 31
2.1.1 Normed Vector Spaces 32
2.1.2 Inner Product Spaces 33
2.2 Stability of a Numerical Method 35
2.2.1 Stability in the Maximum Norm 35
2.2.2 Stability in the Mean-square Sense 39
2.3 Scheme Consistency 42
2.3.1 Consistency of the Three-point FD Scheme 42
2.3.2 Consistency of the P 1 FE Scheme 44
2.4 Convergence of the Discretisation Methods 48
2.4.1 Convergence of the Three-point FDM 49
2.4.2 Convergence of the P 1 FEM 50
2.4.3 Remarks on the Convergence of the FVM 52
2.4.4 Example 2.1: Sensitivity Study of Three Equivalent Methods 54
2.5 Exercises 59
3 Time-dependent Boundary Value Problems 61
3.1 Numerical Solution of the Heat Equation 64
3.1.1 Implicit Time Discretisation 65
3.1.2 Explicit Time Discretisation 66
3.1.3 Example 3.1: Numerical Behaviour of the Forward Euler Scheme 68
3.2 Numerical Solution of the Transport Equation 70
3.2.1 Natural Schemes 70
3.2.2 The Lax Scheme 72
3.2.3 Upwind Schemes 72
3.2.4 Extensions to the FVM and the FEM 73
3.3 Stability of the Numerical Models 76
3.3.1 Schemes for the Heat Equation 77
3.3.2 The Lax Scheme for the Transport Equation 79
3.4 Consistency and Convergence Results 81
3.4.1 Euler Schemes for the Heat Equation 81
3.4.2 Schemes for the Transport Equation 84
3.5 Complements on the Equation of the Vibrating String (VSE) 85
3.5.1 The Lax Scheme to Solve the VS First-order System 85
3.5.2 Example 3.2: Numerical Study of Schemes for the VS First-order System 86
3.5.3 A Natural Explicit Scheme for the VSE 87
3.6 Exercises 90
4 Methods for Two-dimensional Problems 92
4.1 The Poisson Equation 93
4.2 The Five-point FDM 95
4.2.1 Framework and Method Description 95
4.2.2 A Few Words on Possible Extensions 98
4.3 The P 1 FEM 100
4.3.1 Green's Identities 100
4.3.2 The Standard Galerkin Variational Formulation 103
4.3.3 Method Description 104
4.3.4 Implementation Aspects 110
4.3.5 The Master Element Technique 115
4.3.6 Application to Linear Elasticity 117
4.4 Basic FVM 121
4.4.1 The Vertex-centred FVM: Equivalence with the P 1 FEM 122
4.4.2 The Cell-centred FVM: Focus on Flux Computations 126
4.5 SLAE Resolution 138
4.5.1 Example 4.1: A Crout Solver for Banded Matrices 140
4.5.2 Example 4.2: Iterative Solution of Equivalent FD-FE-FV SLAEs 143
4.6 Exercises 147
5 Analyses in Two Space Variables 149
5.1 Methods for the Poisson Equation 150
5.1.1 Convergence of the Five-point FDM 150
5.1.2 Convergence of the P 1 FEM 153
5.1.3 Example 5.1: Solving the Poisson Equation with Neumann Boundary Conditions 164
5.1.4 Example 5.2: Convergence of the P 1 FEM to Non-smooth Solutions 165
5.1.5 Convergence of the FVM 168
5.1.6 Example 5.3: Triangle-centred FVM versus RT0 Mixed FEM 187
5.2 Time Integration Schemes for the Heat Equation 192
5.2.1 Pointwise Convergence of Five-point FD Schemes 193
5.2.2 Convergence of P 1 FE Schemes in the Mean-square Sense 196
5.2.3 Pointwise Behaviour of FE and FV Schemes: An Overview 204
5.3 Exercises 205
6 Extensions 210
6.1 Lagrange FEM of Degree Greater than One 211
6.1.1 The P k FEM in One-dimension Space for k >1 211
6.1.2 A FEM for Quadrilateral Meshes 217
6.1.3 Piecewise Quadratic FEs in Two Space Variables 223
6.1.4 The Case of Curved Domains 225
6.1.5 Example 6.1: P 2-FE Solution of the Equation u - Δu = f 231
6.1.6 More about Implementation in Two-dimensional Space 234
6.2 Extensions to the Three-dimensional Case 240
6.2.1 Methods for Rectangular Domains 241
6.2.2 Tetrahedron-based Methods 245
6.2.3 Implementation Aspects 249
6.2.4 Example 6.2: A MATLAB Code for Three-dimensional FE Computations 252
6.3 Exercises 258
7 Miscellaneous Complements 261
7.1 Numerical Solution of Biharmonic Equations in Rectangles 261
7.1.1 Model Fourth-order Elliptic PDEs 262
7.1.2 The 13-point FD Scheme 263
7.1.3 Hermite FEM in Intervals and Rectangles 265
7.2 The Advection-Diffusion Equation 272
7.2.1 A Model One-Dimensional Equation 272
7.2.2 Overcoming the Main Difficulties with the FDM 274
7.2.3 Example 7.1: Numerical Study of the Upwind FD Scheme 277
7.2.4 The SUPG Formulation 278
7.2.5 Example 7.2: Numerics of the SUPG Formulation for the P 1 FEM 281
7.2.6 An Upwind FV Scheme 282
7.2.7 A FE Scheme for the Time-Dependent Problem 286
7.2.8 Example 7.3: Numerical Study of the Weighted Mass FE Scheme 292
7.3 Basics of a Posteriori Error Estimates and Adaptivity 294
7.3.1 A Posteriori Error Estimates 295
7.3.2 Mesh Adaptivity: h, p and h-p Methods 298
7.4 A Word about Non-linear PDEs 300
7.4.1 Example 7.4: Solving Non-linear Two-point Boundary Value Problems 301
7.4.2 Example 7.5: A Quasi-explicit Method for the Navier-Stokes Equations 305
7.5 Exercises 309
Appendix 311
References 320
Index 331
