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Incompressible Flow and the Finite Element Method, 2 Volume Set




Incompressible Flow and the Finite Element Method, 2 Volume Set

P. M. Gresho, R. L. Sani

ISBN: 978-0-471-49268-9 April 2000 1112 Pages


This comprehensive reference work covers all the important details regarding the application of the finite element method to incompressible flows. It addresses the theoretical background and the detailed development of appropriate numerical methods applied to the solution of a wide range of incompressible flows, beginning with extensive coverage of the advection-diffusion equation in volume one. For both this equation and the equations of principal interest - the Navier-Stokes equations, covered in detail in volume two - detailed 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.
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, this book is an essential reference tool for all researchers, students and applied scientists in incompressible fluid mechanics.
1. Glossary of Abbreviations

2. Preface and Introduction

3. The Navier-Stokes Equations

Notational Introduction

The Continuum Equations

Alternate Forms of the Viscous Term

Alternate Forms of the Non-Linear Term

Derived Equations

Alternate Statements of the NS Equations

Special Cases of Interest

Boundary Conditions

Initial Conditions (and Well-Posedness)

Interim Summary

Global Conservation Laws

Weak Form of the PDEhod

Variational Principles for Potential and Stokes Flow

Solution Methods for the Semi_Discretized Time-Dependent (and Steady) Equations

Aliasing and Aliasing Instability, Linear and Non-Linear

A New Look at Two Old Finite Difference Methods

Numerical Example

Closure: Some Additional Remarks on the Pressure

4. Derived Quantities


Two Dimensions

Three Dimensions

Appendix 1 Some Element Matrix

Appendix 2 Further Comparison of Finite Elements and Finite Volumes

Appendix 3 Projections, Orthagonal and Not and Projection Methods


Author Index

Subject Index
"The book covers the subject in a complete and excellent way, and is most recommended to everybody in this research area. Moreover, the book is written in an interesting, well-organized and very readable way, and approaches the various subjects both by "looking from above" and by going into the details. Sometimes, it also touches on some slightly controversial subjects, which adds flaour to the book." --Dan Givoli, Editor, IACMM Newsletter