Numerische Mathematik

, Volume 107, Issue 4, pp 533–557

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

• Ramon Codina
Article

Abstract

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.

Mathematics Subject Classification (2000)

35Q30 65M12 65M60

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