In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.
Keywords: L2 initial data; Navier–Stokes equations; finite element method; uniform error estimate.