# Incompressible Flow and the Finite Element Method, Volume 1. Advection–Diffusion and Isothermal Laminar Flow

• ID: 2181217
• Book
• 480 Pages
• John Wiley and Sons Ltd
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This comprehensive two volume reference work is devoted to the important details regarding the application of the finite element method to incompressible flows, addressing the theoretical background and the detailed development of appropriate numerical methods applied to their solution. Volume One provides extensive coverage of the prototypical fluid mechanics equation: the advection–diffusion equation. In addition, 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. Also addressed are explanations of how to properly march the time–dependent equations using smart implicit methods. Boundary and initial conditions, so important in applications, are thoroughly described and discussed, including well–posedness. The important role played by the pressure, so confusing in the past, is carefully explained. Together, this two volume work 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
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Preface

Glossary of Abbreviations

1. Introduction

1.1 Introduction

1.2 Incompressible Flow

1.3 The Finite Element Method

1.4 Incompressible Flow and the Finite Element Method

1.5 Overview of this Volume

1.6 Some Subjective Discussion

1.7 Why Finite Elements? Why Not Finite Volumes?

2.1 The Continuum Equation

2.2 The Finite Element Equations /

Discretization of the Weak Form

2.3 Some Semi–Discrete Equations

2.4 Open Boundary Conditions (OBC′s)

2.5 Some Non–Galerkin Results

2.6 Dispersion, Dissipation, Phase Speed, Group Velocity, Mesh Design, and Wiggles

2.7 Time Integration

Appendix 1 Some Element Matrices

A.1.2 One–Dimensional Element Matrices

A.1.3 Two–Dimensional Element Matrices

A.1.4 Two Dimensional Control Volume Finite Element Matrices

Appendix 2 Further Comparison of Finite Elements and Finite Volumes

A.2.1 Introduction

A.2.2 Viewpoint One

A.2.3 Viewpoint Two

Appendix 3 Scalar Projections, Orthogonal and Not and

Projection Methods

A.3.1 Introduction

A.3.2 Scalar Projections

References

Author Index

Subject Index
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