We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier--Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretised in conforming spaces, whose the compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor-Hood finite element.

翻译：我们证明了时间依赖不可压缩纳维-斯托克斯方程增量投影数值格式的收敛性，且无需对弱解施加任何正则性假设。速度和压力在相容空间中离散化，其相容性通过一个正则函数插值器的存在性得以保证，该插值器可保持近似的无散度性质。基于先验估计，我们得到了离散逼近解的存在唯一性。随后，利用适用于时间平移估计的Lions类引理，证明了紧致性性质。进而可以证明近似解收敛至问题的弱解。本文详细给出了最低阶泰勒-胡德有限元情形下插值器的构造过程。