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

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

Spis treści:
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. The Advection–Diffusion Equation
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, Grou p Velocity, Mesh Design, and Wiggles
2.7 Time Integration
2.8 Additional Numerical Examples
Appendix 1 Some Element Matrices
A.1.1 Advection Diffusion 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

Okładka tylna:
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