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.
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Badia, S., Codina, R. Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition. Numer. Math. 107, 533–557 (2007). https://doi.org/10.1007/s00211-007-0099-5
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DOI: https://doi.org/10.1007/s00211-007-0099-5