##
**Implementation of finite element methods for Navier-Stokes equations.**
*(English)*
Zbl 0475.76036

Springer Series in Computational Physics. New York - Heidelberg - Berlin: Springer-Verlag. vii, 161 p. $ 33.60 (1981).

The book is an excellent monograph devoted to finite element methods for Navier-Stokes equations, which doesn’t require any knowledge on structural mechanics. The mathematical aspects of the finite elements are extensively discussed in the book.{

}Thus, the author included in the first chapter several standard finite element examples and the basis ideas of mixed finite element approaches for elliptic equations. The weak formulations of the differential problems provide a mathematical sound background which gives a rigorously scientific character to the book.{

}The second chapter is dedicated to the upwind finite element schemes. Some examples justified their hydrodynamic origin such as: modified weighted residual, reduced integration of the advection term, computation of directional derivatives at the nodes, discontinuous finite elements and mixed interpolation, the method of characteristic in finite elements and perturbation of adjective term. The chapter ends with some numerical examples which emphasize the importance of the upwind schemes in fluid mechanics.{

}The next chapter is the most extensive. It deals with various formulations available for dealing with numerical solutions for the Stokes equations. They are presented as follows: velocity-pressure formulations with discontinuous approximation of the pressure and with continuous approximation of the pressure and velocity; velocity-pressure-velocity formulations: discontinuous approximation of pressure and velocity; vorticity stream-function formulation: decompositions of biharmonic problem. The use of different types of elements (triangle and quadrangle, elements of the first and second order) corresponding to each of these formulations is discussed. In each case the error estimate is given.{

}The chapter dedicated to the Navier-Stokes equations follows the same lines as the previous one. Corresponding formulation for these equations are given. The author’s numerical results based on the velocity pressure formulations: discontinuous approximation of the pressure (with the zero divergence basis) are presented here. It also contains two numerical tests referring to the square wall driven cavity flow and unsteady 2-D flow ground and in an air-intake.{

}The last chapter deals with problems concerning the implementation of the finite element method using the weak formulation of the problems summarily discussed. Two problems are dealt with, that of mesh generation and that of the solutions of the linear and non-linear equations. For the nonlinear equations such techniques as successive approximations, Newton-Raphson algorithm, conjugate gradient method etc. are applied.{

} The book is accompanied by three annexes in which observations from chapters 1 and 3 are demonstrated in detail. References contain more than 300 entries.

}Thus, the author included in the first chapter several standard finite element examples and the basis ideas of mixed finite element approaches for elliptic equations. The weak formulations of the differential problems provide a mathematical sound background which gives a rigorously scientific character to the book.{

}The second chapter is dedicated to the upwind finite element schemes. Some examples justified their hydrodynamic origin such as: modified weighted residual, reduced integration of the advection term, computation of directional derivatives at the nodes, discontinuous finite elements and mixed interpolation, the method of characteristic in finite elements and perturbation of adjective term. The chapter ends with some numerical examples which emphasize the importance of the upwind schemes in fluid mechanics.{

}The next chapter is the most extensive. It deals with various formulations available for dealing with numerical solutions for the Stokes equations. They are presented as follows: velocity-pressure formulations with discontinuous approximation of the pressure and with continuous approximation of the pressure and velocity; velocity-pressure-velocity formulations: discontinuous approximation of pressure and velocity; vorticity stream-function formulation: decompositions of biharmonic problem. The use of different types of elements (triangle and quadrangle, elements of the first and second order) corresponding to each of these formulations is discussed. In each case the error estimate is given.{

}The chapter dedicated to the Navier-Stokes equations follows the same lines as the previous one. Corresponding formulation for these equations are given. The author’s numerical results based on the velocity pressure formulations: discontinuous approximation of the pressure (with the zero divergence basis) are presented here. It also contains two numerical tests referring to the square wall driven cavity flow and unsteady 2-D flow ground and in an air-intake.{

}The last chapter deals with problems concerning the implementation of the finite element method using the weak formulation of the problems summarily discussed. Two problems are dealt with, that of mesh generation and that of the solutions of the linear and non-linear equations. For the nonlinear equations such techniques as successive approximations, Newton-Raphson algorithm, conjugate gradient method etc. are applied.{

} The book is accompanied by three annexes in which observations from chapters 1 and 3 are demonstrated in detail. References contain more than 300 entries.

Reviewer: Titus Petrila (Cluj-Napoca)

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

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

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

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65Z05 | Applications to the sciences |

65M25 | Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |