Navier-Sotkes equations: Theory and approximation.

*(English)*Zbl 0921.76040
Ciarlet, P. G. (ed.) et al., Numerical methods for solids (Part 3). Numerical methods for fluids (Part 1). Amsterdam: Elsevier (ISBN 0-444-82569-X). Handbook of Numerical Analysis 6, 503-688 (1998).

The main goal of this article, which is practically a self-contained book, is to present some selected results on the mathematical theory and numerical approximations of the Navier-Stokes equations for incompressible fluids.

The material is divided into four chapters. Chapter I “Elements of mathematical theory” contains a large number of results on the functional setting of equations, on the existence, uniqueness and regularity of solutions, on the exponential decay of Fourier coefficients in the space-periodic case, long-time behaviour and attractors. The results are presented in a way accessible for numerical analysis, and serve as a theoretical base in the text chapters.

Chapter II “Discretization of the Navier-Stokes equations” examines both time and space discretizations by using an implicit Euler scheme, a first-order semi-implicit scheme, and Crank-Nicolson and Adams-Bashforth schemes. Using mainly energy methods, the authors investigate stability, convergence, and give error estimates for the above schemes.

Chapters III “The incompressibility condition and the computation of laminar flows” and IV “Multi-level methods and the simulation of turbulent flows” are devoted to numerical simulation of the Navier-Stokes equations. A number of methods are described to fulfil the incompressibility condition (projection methods, Uzawa algorithm), and spectral Fourier and finite element methods are shown in the application to the simulation of turbulent flows.

Therefore, this text represents an excellent survey of the current research in the Navier-Stokes equations and can be interesting not only to the particular specialists but also to the fluid mechanicists in general.

For the entire collection see [Zbl 0905.00032].

The material is divided into four chapters. Chapter I “Elements of mathematical theory” contains a large number of results on the functional setting of equations, on the existence, uniqueness and regularity of solutions, on the exponential decay of Fourier coefficients in the space-periodic case, long-time behaviour and attractors. The results are presented in a way accessible for numerical analysis, and serve as a theoretical base in the text chapters.

Chapter II “Discretization of the Navier-Stokes equations” examines both time and space discretizations by using an implicit Euler scheme, a first-order semi-implicit scheme, and Crank-Nicolson and Adams-Bashforth schemes. Using mainly energy methods, the authors investigate stability, convergence, and give error estimates for the above schemes.

Chapters III “The incompressibility condition and the computation of laminar flows” and IV “Multi-level methods and the simulation of turbulent flows” are devoted to numerical simulation of the Navier-Stokes equations. A number of methods are described to fulfil the incompressibility condition (projection methods, Uzawa algorithm), and spectral Fourier and finite element methods are shown in the application to the simulation of turbulent flows.

Therefore, this text represents an excellent survey of the current research in the Navier-Stokes equations and can be interesting not only to the particular specialists but also to the fluid mechanicists in general.

For the entire collection see [Zbl 0905.00032].

Reviewer: O.Titow (Berlin)

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

35Q30 | Navier-Stokes equations |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |