On convergence of Chorin’s projection method to a Leray–Hopf weak solution

Abstract

The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions 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 a 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.