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CONVERGENCE OF THE FULLY DISCRETE INCREMENTAL PROJECTION SCHEME FOR INCOMPRESSIBLE FLOWS

Abstract : The present paper addresses the convergence of a first order in time incremental projection scheme for the time-dependent incompressible Navier-Stokes equations to a weak solution, without any assumption of existence or regularity assumptions on the exact solution. We prove the convergence of the approximate solutions obtained by the semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield the existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.
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https://hal.archives-ouvertes.fr/hal-03727713
Contributor : Raphaele Herbin Connect in order to contact the contributor
Submitted on : Tuesday, July 19, 2022 - 3:15:46 PM
Last modification on : Friday, September 30, 2022 - 9:44:08 AM

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  • HAL Id : hal-03727713, version 1
  • ARXIV : 2207.09695

Citation

Thierry Gallouët, Raphaèle Herbin, Jean-Claude Latché, David Maltese. CONVERGENCE OF THE FULLY DISCRETE INCREMENTAL PROJECTION SCHEME FOR INCOMPRESSIBLE FLOWS. 2022. ⟨hal-03727713⟩

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