Finite element methods for the numerical simulation of incompressible viscous flow. Introduction to the control of the Navier-Stokes equations. (English) Zbl 0751.76046

Glowinski, Roland

Vortex dynamics and vortex methods, Proc. 21st AMS-SIAM Semin., Seattle/WA (USA) 1990, Lect. Appl. Math. 28, 219-301 (1991).

Summary: [For the entire collection see Zbl 0742.00057.]
In this article we discuss the solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow, by numerical methods combining operator splitting for the time discretization and finite elements for the space discretization. The discussion includes the description of conjugate gradient algorithms, which are used to solve the advection-diffusion and Stokes type problems produced at each time step by the operator splitting methods. This paper includes an introduction to control problems for the Navier-Stokes equations, which is a topic of growing importance. The results of numerical experiments are presented; from these results, which include flow simulation at Reynolds numbers of the order of \(10^ 4\), it appears clearly that the methodology discussed here is robust and accurate.

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
76D05	Navier-Stokes equations for incompressible viscous fluids
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

operator splitting; conjugate gradient algorithms; advection-diffusion; Stokes type problems