Finite element methods for Navier-Stokes equations. Theory and algorithms. (Extended version of the 1979 publ.). (English) Zbl 0585.65077

Girault, Vivette; Raviart, Pierre-Arnaud

Springer Series in Computational Mathematics, 5. Berlin etc.: Springer-Verlag. X, 374 p. (1986).

The present work is an extended version of a previous text on finite element approximation of the Navier-Stokes equations [Lect. Notes Math. 749 (1979; Zbl 0413.65081)]. It is concerned with theory and application of finite element methods to the stationary Navier-Stokes equations for incompressible fluid flows. The authors restrict themselves to inner problems and provide a fairly comprehensive and exhaustive up to date treatment of this field. Emphasis is given on the mathematical foundation of finite element methods. As to implementation and numerical results the authors refer to the complementary work of F. Thomasset [Implementation of finite element methods for Navier-Stokes equations (1981; Zbl 0475.76036)] and of R. Peyret and T. Taylor [Computational methods for fluid flow (1983; Zbl 0514.76001)].
The book is devided into four chapters and a technical appendix: Chapter I is devoted to theoretical aspects of the Stokes equations including a study of related function spaces and variational formulations of the underlying problem. Chapter II is devoted to finite element approximations to the Stokes problem in the original variables (velocity and pressure). Here, the most popular mixed schemes are presented. In chapter III various finite element methods are presented that rest upon other variables such as stream function, vector potential and vorticity. Finally, chapter IV is concerned with theory and approximation of the full Navier-Stokes equations where the results of the two preceding chapters are systematically extended to the nonlinear equations.

Reviewer: W.Velte

MSC:

65Z05	Applications to the sciences
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76-02	Research exposition (monographs, survey articles) pertaining to fluid mechanics
35A15	Variational methods applied to PDEs
35G20	Nonlinear higher-order PDEs
35Q30	Navier-Stokes equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65N15	Error bounds for boundary value problems involving PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids
76D07	Stokes and related (Oseen, etc.) flows
76M99	Basic methods in fluid mechanics

Keywords:

finite element methods; inner problems; Stokes problem

Citations:

Zbl 0413.65081; Zbl 0475.76036; Zbl 0514.76001