Finite element solution of the Navier-Stokes equations. (English) Zbl 0801.76043

Fortin, Michel

Iserles, A. (ed.), Acta Numerica 1993. Cambridge: Cambridge University Press. 239-284 (1993).

The rough contents of this important and elegant paper are as follows: 1. Introduction; 2. The finite element method; 3. Presentation of the problem; 4. The Stokes problem: incompressibility and pressure; 5. Finite elements for incompressible problems, 6. The \(Q_ 1-P_ 0\) element (or “what might go wrong”); 7. Stabilization techniques; 8. Numerical methods for the discretized problems; 9. Time-dependent problems; 10. Conclusion and references.
The author is an active researcher in the topic of numerical solutions for Navier-Stokes equations, so he realized a keen analysis of some difficulties encountered in solving these equations. The main issue treated is the incompressibility effect. In spite of the fact that the paper is not an exhaustive presentation (no one could have pretended), it is extremely rich in ideas, with clear and precise statements. It is also my feeling that this article represents “a useful guide to the rapidly changing world of computational fluid dynamics” – to quote the last words of the author.
For the entire collection see [Zbl 0777.00039].

Reviewer: C.I.Gheorghiu (Cluj-Napoca)

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
76D05	Navier-Stokes equations for incompressible viscous fluids
76D07	Stokes and related (Oseen, etc.) flows
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

variational formulation; error estimates; stabilization; incompressibility effect