A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations

We present a fractional-step finite element method based on a subgrid model for simulating the time-dependent incompressible Navier–Stokes equations. The method aims to the simulation of high Reynolds number flows and consists of two steps in which the nonlinearity and incompressibility are split into different steps. The first step of this method can be seen as a linearized Burger’s problem where a subgrid model based on an elliptic projection of the velocity into a lower-order finite element space is employed to stabilize the system, and the second step is a Stokes problem. Under mild regularity assumptions on the continuous solution, we obtain the stability of the numerical method, and derive error bound of the approximate velocity, which shows that first-order convergence rate in time and optimal convergence rate in space can be gotten by the method. Numerical experiments verify the theoretical predictions and demonstrate the promise of the proposed method, which show superiority of the proposed method to the compared method in the literature.