Finite Element Approximation of the Nonstationary Navier-Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spatial Discretization

This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier-Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as t → 0 and as t → ∞. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier-Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.