Skip to main content
main-content

Über dieses Buch

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 comprehen­ sive treatment of the most recent developments in that field. To stay within reasonable bounds, we have restricted ourselves to the case of stationary prob­ lems 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 implemen­ tation 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].

Inhaltsverzeichnis

Frontmatter

Chapter I. Mathematical Foundation of the Stokes Problem

Abstract
This paragraph contains a short survey on the Dirichlet’s and Neumann’s problems for the harmonic and biharmonic operators.
Vivette Girault, Pierre-Arnaud Raviart

Chapter II. Numerical Solution of the Stokes Problem in the Primitive Variables

Abstract
The abstract problem discussed in Chapter I, § 4 lends itself readily to a straight-forward approximation that converges under reasonable assumptions with an error proportional to the approximation error of the spaces involved. When applied to the Stokes problem, this approach yields a conforming approximation of the velocity and pressure, although the approximate velocity field is (in general) not exactly divergence-free. The wide range of finite element methods developped in the remainder of the chapter are all founded on the material of this paragraph. Non-conforming methods can also be put into this framework (cf. Zine [85]) but for the sake of conciseness we have skipped them entirely.
Vivette Girault, Pierre-Arnaud Raviart

Chapter III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem

Abstract
In this paragraph, we concentrate again upon the abstract problem studied in Chapter I § 4, but we put it into a weaker setting leading to a (generally) different mixed formulation. The mixed approximation derived from this formulation will give rise to the important class of exactly incompressible methods to solve the Stokes and Navier-Stokes equations.
Vivette Girault, Pierre-Arnaud Raviart

Chapter IV. Theory and Approximation of the Navier-Stokes Problem

Abstract
In this paragraph, we study a nonlinear generalization of the abstract variational problem analyzed in Paragraph I.4. This family of nonlinear problems contains in particular the Navier-Stokes problem.
Vivette Girault, Pierre-Arnaud Raviart

Backmatter

Weitere Informationen