Incompressible Flow and the Finite Element Method, Volume 1

Volume 1

Preface xv

Glossary of Abbreviations xix

1 Introduction 1

1.1 Introduction 1

1.2 Incompressible Flow 3

1.3 The Finite Element Method 6

1.4 Incompressible Flow and the Finite Element Method 11

1.5 Overview of this Volume 12

1.6 Some Subjective Discussion 16

1.7 Why Finite Elements? Why Not Finite Volumes? 17

2 The Advection-Diffusion Equation 21

2.1 The Continuum Equation 21

2.2 The Finite Element Equations/Discretization of the Weak Form 35

2.3 Same Semi-Discrete Equations 56

2.4 Open Boundary Conditions (OBC’s) 91

2.5 Same Non-Galerkin Results 105

2.6 Dispersion, Dissipation, Phase Speed, Group

2.7 Time Integration 230

2.8 Additional Numerical Examples 342

Appendix 1 Some Element Matrices 357

Appendix 2 Further Comparison of Finite Elements and Finite Volumes 365

Appendix 3 Scalar Projections, Orthogonal and Not—and Projection Methods 379

References 423

Author Index Ai-1

Subject Index Si-1

Volume 2

Glossary of Abbreviations xv

Preface and Introduction xvii

Preface xvii

Introduction xx

Incompressible Flow xxii

The Finite Element Method xxv

Incompressible Flow and the Finite Element Method xxvi

Overview of this Volume xxxi

Some Subjective Discussion xxxv

Why Finite Elements? Why Not Finite Volumes? xxxvi

3 The Navier–Stokes Equations 447

3.1 Notational Introduction 447

3.2 The Continuum Equations (The PDE’s) 450

3.3 Alternate Forms of the Viscous Term 452

3.4 Alternate Forms of the Non-Linear Term 454

3.5 Derived Equations 457

3.6 Alternate Statements of the NS Equations 461

3.7 Special Cases of Interest 463

3.8 Boundary Conditions 470

3.9 Initial Conditions (and Well-Posedness) 487

3.10 Interim Summary 493

3.11 Global Conservation Laws 502

3.12 Weak Forms of the PDE’s/Natural Boundary Conditions (NBC’s) 508

3.13 The Finite Element Equations/Discretization of the Weak Form 528

3.14 A Control Volume Finite Element Method 712

3.15 Variational Principles for Potential and Stokes Flow 716

3.16 Solution Methods for the Semi-Discretized Time-Dependent (and Steady) Equations 729

3.17 Aliasing and Aliasing Instability, Linear and Non-Linear 876

3.18 A New Look al Two Old Finite Difference Methods 880

3.19 Numerical Example-Impulsive Start 884

3.20 Closure: Some Additional Remarks on the Pressure 934

4 Derived Quantities 937

4.1 Introduction 937

4.2 Two Dimensions 938

4.3 Three Dimensions 961

4.3.1 Vorticity 961

4.3.2 Helicity Density 961

Appendix 4 Some More Element Matrices 963

Appendix 5 Vector Projections, Orthogonal and Not—and Projection Methods 967

References 989

Author Index Ai-1

Subject Index Si-1