Before revision: An error estimate is presented for a fully discrete semi-implicit finite element method for the two-dimensional Navier--Stokes equations with L^2 initial data in convex polygonal domains, without extra regularity assumptions or CFL conditions, by utilizing the smoothing properties of the Navier--Stokes equations, an appropriate duality argument, and the smallness of the numerical solution in the discrete L^2(0,t_m;H^1) norm when t_m is smaller than some constant.
After revision: First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier--Stokes equations with L^2 initial data in convex polygonal domains, without extra regularity assumptions or CFL conditions, by utilizing the smoothing properties of the Navier--Stokes equations, an appropriate duality argument, and the smallness of the numerical solution in the discrete L^2(0,t_m;H^1) norm when t_m is smaller than some constant.