  Kuroki and Soga [Numer. Math. 2020] proved that a version of Chorin's fully
discrete projection method, originally introduced by A. J. Chorin [Math. Comp.
1969], is unconditionally solvable and convergent within an arbitrary fixed
time interval to a Leray-Hopf weak solution of the incompressible Navier-Stokes
equations on a bounded domain with an arbitrary external force. This paper is a
continuation of Kuroki-Soga's work. We show time-global solvability and
convergence of our scheme; $L^2$-error estimates for the scheme in the class of
smooth exact solutions; application of the scheme to the problem with a
time-periodic external force to investigate time-periodic (Leray-Hopf weak)
solutions, long-time behaviors, error estimates, etc.
