Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

by Vivette Girault, Pierre-Arnaud Raviart

Overview

The material covered by this book has been taught by one of the authors in a post-graduate course on Numerical Analysis at the University Pierre et Marie Curie of Paris. It is an extended version of a previous text (cf. Girault & Raviart [32J) published in 1979 by Springer-Verlag in its series: Lecture Notes in Mathematics. In the last decade, many engineers and mathematicians have concentrated their efforts on the finite element solution of the Navier-Stokes equations for incompressible flows. The purpose of this book is to provide a fairly comprehensive treatment of the most recent developments in that field. To stay within reasonable bounds, we have restricted ourselves to the case of stationary problems although the time-dependent problems are of fundamental importance. This topic is currently evolving rapidly and we feel that it deserves to be covered by another specialized monograph. We have tried, to the best of our ability, to present a fairly exhaustive treatment of the finite element methods for inner flows. On the other hand however, we have entirely left out the subject of exterior problems which involve radically different techniques, both from a theoretical and from a practical point of view. Also, we have neither discussed the implementation of the finite element methods presented by this book, nor given any explicit numerical result. This field is extensively covered by Peyret & Taylor [64J and Thomasset [82].

Product Details

ISBN-13:	9783642648885
Publisher:	Springer Berlin Heidelberg
Publication date:	10/01/2011
Series:	Springer Series in Computational Mathematics , #5
Edition description:	Softcover reprint of the original 1st ed. 1986
Pages:	376
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

I. Mathematical Foundation of the Stokes Problem.- § 1. Generalities on Some Elliptic Boundary Value Problems.- §2. Function Spaces for the Stokes Problem.- § 3. A Decomposition of Vector Fields.- §4. Analysis of an Abstract Variational Problem.- §5. The Stokes Equations.- Appendix A. Results of Standard Finite Element Approximation.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- §1. General Approximation.- § 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- § 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- §4. Continuous Approximation of the Pressure.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- §1. Mixed Approximation of an Abstract Problem.- §2. The “Stream Function-Vorticity-Pressure” Method for the Stokes Problem in Two Dimensions.- § 3. Further Topics on the “Stream Function-Vorticity-Pressure” Scheme.- § 4. A “Stream Function-Gradient of Velocity Tensor” Method in Two Dimensions.- § 5. A “Vector Potential-Vorticity” Scheme in Three Dimensions.- IV. Theory and Approximation of the Navier-Stokes Problem.- § 1. A Class of Nonlinear Problems.- §2. Theory of the Steady-State Navier-Stokes Equations.- § 3. Approximation of Branches of Nonsingular Solutions.- §4. Numerical Analysis of Centered Finite Element Schemes.- § 5. Numerical Analysis of Upwind Schemes.- §6. Numerical Algorithms.- References.- Index of Mathematical Symbols.