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：我们证明了随时间变化的不可压缩纳维-斯托克斯方程的一种增量投影数值格式的收敛性，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，其兼容性通过一个保持近似无散特性的正则函数插值器得到保证。借助先验估计，我们获得了离散近似解的存在唯一性。随后，基于用于时间平移估计的Lions型引理，证明了紧致性性质。由此可以证明近似解收敛到问题的一个弱解。文中详细介绍了最低阶Taylor-Hood有限元情形下插值器的构造方法。