The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data

This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data $u_0\in H^\alpha$, i.e., the time step condition is $\tau\leq C_0$ in the case of $\alpha=2$, $\tau\vert\log h\vert\leq C_0$ in the case of $\alpha=1$ and $\tau h^{-2}\leq C_0$ in the case of $\alpha=0$ for mesh size $h$ and some positive constant $C_0$. We provide the $H^2$-stability of the scheme under the stability condition with $\alpha=0,1,2$ and obtain the optimal $H^1-L^2$ error estimate of the numerical velocity and the optimal $L^2$ error estimate of the numerical pressure under the stability condition with $\alpha=1,2$.