Navier-Stokes Equations : Theory and Numerical Analysis

by Temam, Roger

• ISBN13: 9780821827376
• ISBN10: 0821827375
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Amer Mathematical Society

Summary

The book presents a systematic treatment of results on the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. Considered are the linearized stationary case, the nonlinear stationary case, and the full nonlinear time-dependent case. The relevant mathematical tools are introduced at each stage. The new material in this book is Appendix III, reproducing a survey article written in 1998. This appendix contains a few aspects not addressed in the earlier editions, in particular a short derivation of the Navier-Stokes equations from the basic conservation principles in continuum mechanics, further historical perspectives, and indications on new developments in the area. The appendix also surveys some aspects of the related Euler equations and the compressible Navier-Stokes equations. Readers are advised to peruse this appendix before reading the core of the book. This book presents basic results on the theory of Navier-Stokes equations and, as such, continues to serve as a comprehensive reference source on the topic.

Table of Contents

Preface to the AMS Chelsea edition	ix
Preface to the third (revised) edition	xi
Foreword	xiii
The Steady-State Stokes Equations	1	(104)
Introduction	1	(1)
Some function spaces	1	(14)
Existence and uniqueness for the Stokes equations	15	(13)
Discretization of the Stokes equations (I)	28	(17)
Discretization of Stokes Equations (II)	45	(46)
Numerical algorithms	91	(7)
The penalty method	98	(7)
Steady-State Navier-Stokes Equations	105	(62)
Introduction	105	(1)
Existence and uniqueness theorems	105	(16)
Discrete inequalities and compactness theorems	121	(13)
Approximation of the stationary Navier-Stokes equations	134	(16)
Bifurcation theory and non-uniqueness results	150	(17)
The Evolution Navier-Stokes Equation	167	(144)
Introduction	167	(1)
The linear case	167	(15)
Compactness theorems	182	(7)
Existence and uniqueness theorems (n ≤ 4)	189	(27)
Alternate proof of existence by semi-discretization	216	(8)
Discretization of the Navier-Stokes equations: General stability and convergence theorems	224	(22)
Discretization of the Navier-Stokes equations: Application of the general results	246	(21)
Approximation of the Navier-Stokes equations by the projection method	267	(20)
Approximation of the Navier-Stokes equations by the artificial compressibility method	287	(24)
Appendix I. Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations	311	(10)
1. Functional properties of the curl operator	311	(7)
2. Application to the non-homogeneous steady-state Navier-Stokes equations	318	(3)
Appendix II. Implementation of Non-conforming Linear Finite Elements (Approximation APX5-Two-dimensional Case)	321	(16)
F. Thomasset
0. Test problems	321	(1)
1. The triangulation	321	(5)
2. The nodes	326	(1)
3. Computation of the basis function on a given triangle	327	(2)
4. Solution of the Stokes problem	329	(4)
5. Solution of Navier-Stokes equations	333	(4)
Appendix III. Some Development on Navier-Stokes Equations in the Second Half of the 20th Century	337	(44)
Introduction	337	(2)
Part I: The incompressible Navier-Stokes equations	339	(1)
1. Existence, uniqueness and regularity of solutions	339	(14)
2. Attractors and turbulence	348	(5)
Part II: Other problems, other equations	353	(1)
3. Compressible and inviscid flows	353	(5)
4. Some other problems and equations	358	(23)
Bibliography to Appendix III	363	(18)
Comments	381	(6)
Chapter 1	381	(1)
Chapter 2	382	(1)
Chapter 3	383	(4)
Additional comments to the third (revised) edition	387	(2)
Bibliography (to the core of the book)	389	(18)
Index	407