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 twodimensional (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 u0 ∈ Hα, i.e., the time step condition is τ ≤ C0 in the case of α = 2, τ | log h| ≤ C0 in the case of α = 1 and τh−2 ≤ C0 in the case of α = 0 for mesh size h and some positive constant C0. We provide the H2-stability of the scheme under the stability condition with α = 0, 1, 2 and obtain the optimal H1−L2 error estimate of the numerical velocity and the optimal L2 error estimate of the numerical pressure under the stability condition with α = 1, 2.