On convergence of Chorin's projection method to a Leray-Hopf weak solution

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.