  The projection method to solve the incompressible Navier-Stokes equations was
first studied by Chorin [Math. Comp., 1969] in the framework of a finite
difference method and Temam [Arch. Rational Mech. and Anal., 1969] in the
framework of a finite element method. Chorin showed convergence of
approximation and its error estimates in problems with the periodic boundary
condition assuming existence of a $C^5$-solution, while Temam demonstrated an
abstract argument to obtain a Leray-Hopf weak solution in problems on a bounded
domain with the no-slip boundary condition. In the present paper, the authors
extend Chorin's result with full details to obtain convergent finite difference
approximation of a Leray-Hopf weak solution to the incompressible Navier-Stokes
equations on an arbitrary bounded Lipschitz domain of $\mathbb{R}^3$ with the
no-slip boundary condition and an external force. We prove unconditional
solvability of our implicit scheme and strong $L^2$-convergence (up to
subsequence) under the scaling condition $ h^{3-\alpha}\le\tau$ (no upper bound
is necessary), where $h,\tau$ are space, time discretization parameters,
respectively, and $\alpha\in(0,2]$ is any fixed constant. The results contain a
compactness method based on a new interpolation inequality for step functions.
