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

• M. O. Bristeau
• R. Glowinski
• B. Mantel
• J. Periaux
• P. Perrier
• O. Pironneau
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 771)

## Abstract

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:
1. A mixed finite element approximation acting on a pressure-velocity formulation of the problem,

2. A time discretization by finite differences for the unsteady problem,

3. 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 $$- \varepsilon \Delta u + \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{\beta } \cdot \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{\nabla } u = f$$ on Ω, u=0 on ∂Ω (Ω : bounded domain of ℝ2), but we plan to apply it to the solution of Navier-Stokes problems.

## Keywords

Stokes Problem Finite Element Approximation Conjugate Gradient Algorithm Mixed Finite Element Approximate Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• M. O. Bristeau
• 1
• R. Glowinski
• 2
• B. Mantel
• 3
• J. Periaux
• 3
• P. Perrier
• 3
• O. Pironneau
• 4
1. 1.Iria-LaboriaLe ChesnayFrance
2. 2.Université Paris VI, L.A. 189Paris
3. 3.AMD/BA, 78 Quai CarnotSt-CloudFrance
4. 4.Université Paris-NordSt-DenisFrance