Finite element methods for incompressible viscous flow. (English) Zbl 1040.76001

Glowinski, Roland

Ciarlet, P. G. (ed.) et al., Numerical methods for fluids (Part 3). Amsterdam: North-Holland (ISBN 0-444-51224-1/hbk). Handb. Numer. Anal. 9, 1176 p. (2003).

The main purpose of this book is to present a detailed and self-contained description of solution methods for Navier-Stokes equations modeling the unsteady flow of incompressible Newtonian viscous fluids. The methods described in this volume rely on variational principles with strong emphasis on low order finite element approximations (typically piecewise linear or piecewise quadratic), since they are fairly easy to implement, well suited to complicated geometries and lead to reasonably well-conditioned finite-dimensional problems. The coverage contains the following chapters:
Chapter I. The Navier-Stokes equations for incompressible viscous fluids. Chapter II. A family of operator-splitting methods for initial value problems. Application to the Navier-Stokes equations. Chapter III. Iterative solution of the advection-diffusion subproblems. Chapter IV. Iterative solution of the Stokes subproblems. Chapter V. Finite element approximation of the Navier-Stokes equations. Chapter VI. Treatment of the advection by a wave-like equation method and by backward methods of characteristics. Chapter VII. On \(L^2\)-projection methods for the numerical treatment of the incompressibility. Chapter VIII. Fictitious domain methods for incompressible viscous flow: Application to particulate flow. Chapter IX. Numerical experiments. Chapter X. Complements: From stream function-vorticity to flow control.
Scientists working in the areas of numerical analysis and fluid mechanics will benefit from this book, which may serve both communities as both a reference monograph and a textbook for computational fluid dynamics courses.
For the entire collection see [Zbl 1020.00003].

Reviewer: Felix Kaplanski (Tallinn)

MSC:

76-00	General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to fluid mechanics
76-02	Research exposition (monographs, survey articles) pertaining to fluid mechanics
76M10	Finite element methods applied to problems in fluid mechanics
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

Navier-Stokes equations; variational principles; low-order finite element approximations; operator-splitting methods; wave-like equation method; backward methods of characteristics; projection methods; fictitious domain methods