The present paper addresses the convergence of a first order in time incremental projection scheme for the time-dependent incompressible Navier-Stokes equations to a weak solution, without any assumption of existence or regularity assumptions on the exact solution. We prove the convergence of the approximate solutions obtained by the semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield the existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.
翻译:本文件讨论了时间依赖的压抑性纳维埃-斯托克斯等方程式在时间上先顺序递增预测方案与一个薄弱的解决方案的趋同问题,而没有假定存在,也没有假定确切解决方案是否合乎常态。我们证明半分立办法和完全独立的办法获得的近似解决办法的趋同,在非统一的矩形藻类上采用错开的有限体积办法;首先对近似解决办法作出先验估计,从而产生存在。然后根据这些估计数,根据这些估计数,对离散时间衍生物的翻译作一些估计,拟订契约性论点,以便取得趋同(直至提取一个后继序列),因为半分立办法的时间步骤往往为零,而完全分解办法的空间和时间步骤往往为零;因此,近似解决办法显示会与一个限制功能趋同,而后又通过将这些办法的限度转至持续问题的一个较弱的解决办法。