This book-size article is dedicated to the numerical simulation of unsteady incompressible viscous flow modelled by the Navier-Stokes equations, or by non-Newtonian variants of them. In order to achieve this goal a methodology has been developed based on four key tools. Time discretization by operator-splitting schemes such as Peaceman-Rachford's, Douglas Rachford's, Marchuk-Yanenko's, Strang's symmetrized, and the so-called "theta-scheme" introduced by the author in the mid-1980s. Projection methods (in L2 or H1) for the treatment of the incompressibility condition div u = 0. Treatment of the advection by: either a centered scheme leading to linear or nonlinear advection-diffusion problems solved by least squares/conjugate gradient algorithms, or to a linear wave-like equation well suited to finite element-based solution methods. Space approximation by finite element methods such as Hood-Taylor and Bercovier-Pironneau, which are relatively easy to implement. In addition to the above topics the article contains detailed discussions of conjugate gradient algorithms, least-squares methods for boundary-value problems which are not equivalent to problems of the calculus of variations, Uzawa-type algorithms for the solution of saddle-point problems, embedding/fictitious domain methods for the solution of elliptic and parabolic problems. In fact many computational methods discussed in this article also apply to non-CFD problems although they were mostly designed for the solution of flow problems. Among the topics covered are: the direct numerical simulation of particulate flow; computational methods for flow control; splitting methods for viso-plastic flow a la Bingham; and more. It should also be mentioned that most methods discussed in this article are illustrated by the results of numerical experiments, including the simulation of three-dimensional flow. Due to their modularity the methods described in this article are relatively easy to implement - as is demonstrated by the fact that several practitioners in various institutions have been able to use them ab initio for the solution of complicated flow (and other) problems.

Table of Contents:
Chapter I The Navier-Stokes equations for incompressible viscous fluids: derivation of the Navier-Stokes equations for viscous fluids; initial and boundary conditions; a stream function-vorticity formulation of the Navier-Stokes equations; a brief introduction to Sobolev spaces; variational formulations of the Navier-Stokes equations; a short review of mathematical results for the Navier-Stokes equations. Chapter II A family of operator splitting methods for initial value problems - application to the Navier-Stokes equations: a family of initial value problems; the Peaceman-Rachford method; the Douglas-Rachford method; A-scheme; application to the Navier-Stokes equations. Chapter III Iterative solution of the advection-diffusion subproblems: classical and variational formulations of the advection-diffusion subproblems associated with the operator splitting schemes; linear variational problems in Hilbert spaces; variational methods for the solution of the advection-diffusion problems (13.1) and (13.2); conjugate gradient methods for the solution of minimization problems in Hilbert spaces; least-squares solution of linear and nonlinear problems in Hilbert spaces; least-squares/conjugate gradient solution of problems (13.1) and (13.2). Chapter IV Iterative solution of the Stokes subproblems: mathematical properties of the generalized Stokes problem (GS)1; gradient methods for the Stokes problem; conjugate gradient methods for the Stokes problem (GS)1; iterative solution of the generalized Stokes problem (GS)2; on artificial compressibility methods and further comments. Chapter V Finite element approximation of the Navier-Stokes equations: solution of the Stokes problem with periodic boundary conditions; a Fourier analysis of the numerical instability mechanism; finite element implementation of the scheme (11.5)-(11.8); on the numerical solution of the discrete subproblems. Chapter VI Treatment of the advection by a wave-like equation method and by backward methods of characteristics: more on operator-splitting methods; a wave-like equation method for solving the Navier-Stokes equations; solution of the Navier-Stokes equations by backward methods of characteristics; on the treatment of the advection by upwinding. Chapter VII On L2-projection methods for the numerical treatment of the incompressibility: combining L2-projection methods with operator-splitting schemes la Peaceman-Rachford and Douglas-Rachford, and with the scheme; combining L2-projection methods with operator splitting schemes la Marchuk-Yanenko; numerical experiments. Chapter VIII Fictitious domain methods for incompressible viscous flow - application to particulate flow. Chapter X Complements - from stream function-vorticity to flow control.