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The present paper addresses the convergence of a first order in timeincremental projection scheme for the time-dependent incompressibleNavier-Stokes equations to a weak solution, without any assumption of existenceor regularity assumptions on the exact solution. We prove the convergence ofthe approximate solutions obtained by the semi-discrete scheme and a fullydiscrete scheme using a staggered finite volume scheme on non uniformrectangular meshes. Some first a priori estimates on the approximate solutionsyield the existence. Compactness arguments, relying on these estimates,together with some estimates on the translates of the discrete timederivatives, are then developed to obtain convergence (up to the extraction ofa subsequence), when the time step tends to zero in the semi-discrete schemeand 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 whichis then shown to be a weak solution to the continuous problem by passing to thelimit in these schemes.

Convergence of the fully discrete incremental projection scheme for incompressible flows