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,对于基于时间的不压缩纳维埃-斯托克斯方程式,一个递增的预测数字方案是趋同的,对薄弱的解决方案没有定期的假设。速度和压力在符合的空格中是分解的,这些空格的兼容性由常规功能的内插器来保证,这些正常功能的内插器可以保持大约的不偏差特性。根据先验的估计,我们得到离散近点的存在和独特性。随后,对紧凑性特性进行了证明,在时间翻译估计数时,依靠像狮子一样的利姆马来进行翻译。然后,有可能显示近似的解决办法与问题较弱的解决方案的趋同性。在最低程度的泰勒-哈德有限要素的情况下,对内插器的构造进行了详细描述。