This comprehensive two-volume reference covers the application of the finite element method to incompressible flows in fluid mechanics, addressing the theoretical background and the development of appropriate numerical methods applied to their solution. Volume One provides extensive coverage of the prototypical fluid mechanics equation: the advection-diffusion equation. For both this equation and the equations of principal interest - the Navier-Stokes equations (covered in detail in Volume Two) - a discussion of both the continuous and discrete equations is presented, as well as explanations of how to properly march the time-dependent equations using smart implicit methods. Boundary and initial conditions, so important in applications, are carefully described and discussed, including well-posedness. The important role played by the pressure, so confusing in the past, is carefully explained. The book explains and emphasizes consistency in six areas: * consistent mass matrix * consistent pressure Poisson equation * consistent penalty methods * consistent normal direction * consistent heat flux * consistent forces Fully indexed and referenced, this book is an essential reference tool for all researchers, students and applied scientists in incompressible fluid mechanics.

Table of Contents:
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