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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 ∈ H α , i.e., the time step condition is τ < C 0 in the case of a = 2, τ| log h| ≤ C 0 in the case of a = 1 and Th -2 ≤ C 0 in the case of a = 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 a = 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 a = 1, 2.

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