  A first-order linear fully discrete scheme is studied for the incompressible
time-dependent Navier-Stokes equations in three-dimensional domains. This
scheme, based on an incremental pressure projection method, decouples each
component of the velocity and the pressure, solving in each time step, a linear
convection-diffusion problem for each component of the velocity and a
Poisson-Neumann problem for the pressure.
  Using first-order \emph{inf-sup} stable $C^0$-finite elements, optimal error
estimates of order $O(k+h)$ are deduced without imposing constraints on $h$ and
$k$, the mesh size and the time step, respectively.
  Finally, some numerical results are presented according the theoretical
analysis, and also comparing to other current first-order segregated schemes.
