On optimal convergence rates for higher-order Navier–Stokes approximations. I. Error estimates for the spatial discretization

Abstract

Due to the increasing use of higher-order methods in computational fluid dynamics, the question of optimal approximability of the Navier-Stokes equations under realistic assumptions on the data has become important. It is well known that the regularity customarily hypothesized in the error analysis for parabolic problems cannot be assumed for the Navier-Stokes equations, as it depends on non-local compatibility conditions for the data at time t = 0, which cannot be verified in practice. Taking into account this loss of regularity at t = 0, improved convergence of the order O(min{h (5/2)-δ , h 3 /t (1/4)+δ }), for any δ > 0, is shown locally in time for the spatial discretization of the velocity field by (non-)conforming finite elements of third-order approximability properties. The error estimate itself is proved by energy methods, but it is based on sharp a priori estimates for the Navier-Stokes solution in fractional-order spaces that are derived by semigroup methods and complex interpolation theory and reflect the optimal regularity of the solution as t → 0.