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.

中文翻译：

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关于 Chorin 投影方法对 Leray-Hopf 弱解的收敛性

求解不可压缩 Navier-Stokes 方程的投影方法首先由 Chorin [Math. Comp., 1969] 在有限差分法和 Temam [Arch. 理性机械。和分析，1969] 在有限元方法的框架内。Chorin 在假设存在 $C^5$-解的周期性边界条件问题中展示了近似的收敛性及其误差估计，而 Temam 展示了一个抽象论证，以在有界域上的问题中获得 Leray-Hopf 弱解，其中无滑移边界条件。在本文中，作者扩展了 Chorin' s 结果与完整的细节，以获得在 $\mathbb{R}^3$ 的任意有界 Lipschitz 域上不可压缩 Navier-Stokes 方程的 Leray-Hopf 弱解的收敛有限差分近似，具有无滑移边界条件和外部压力。我们证明了在缩放条件 $ h^{3-\alpha}\le\tau$（不需要上限）下我们的隐式方案和强 $L^2$-收敛（直到子序列）的无条件可解性，其中 $ h,\tau$ 分别是空间、时间离散化参数，$\alpha\in(0,2]$ 是任意固定常数。结果包含基于新的阶跃函数插值不等式的紧致方法。