Finite Element Approximation of the Nonstationary Navier–Stokes Problem III. Smoothing Property and Higher Order Error Estimates for Spatial Discretization

This paper continues our error analysis of finite element Galerkin approximation of the nonstationary Navier–Stokes equations. Optimal order error estimates, both local and global, are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution. These assumptions take into account the loss of regularity at t = 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem.