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) is an element of H-alpha, i.e., the time step condition is tau <= C-0 in the case of alpha = 2, tau|log h| <= C-0 in the case of alpha = 1 and tau h(-2) = 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.