  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.
