A finite element approximation of Navier-Stokes equations for incompressible viscous fluids. Iterative methods of solution

We present in this paper a method for the numerical solution of the steady and unsteady Navier-Stokes equations for incompressible viscous fluids. This method is based on the following techniques:A mixed finite element approximation acting on a pressure-velocity formulation of the problem,A time discretization by finite differences for the unsteady problem,An iterative method using — via a convenient nonlinear least square formulation — a conjugate gradient algorithm with scaling; the scaling makes a fundamental use of an efficient Stokes solver associated to the above mixed finite element approximation. The results of numerical experiments are presented and analyzed. We conclude this paper by an appendix introducing a new upwind finite element approximation; we discuss in this appendix the solution by this new method of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \varepsilon \Delta u + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\beta } \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\nabla } u = f$$\end{document} on Ω, u=0 on ∂Ω (Ω : bounded domain of ℝ2), but we plan to apply it to the solution of Navier-Stokes problems.