Uniform Error Estimates of the Finite Element Method for the Navier–Stokes Equations in R[sup.2] with IL/I[sup.2] Initial Data

In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L[sup.2] 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 H[sup.1]-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 H[sup.1]-norm and the pressure in L[sup.2]-norm.