Error estimates for the spectral Galerkin approximations of the solutions of Navier–Stokes type equations

Abstract

In [8], [9] R. Rautmann has given a systematic development of error estimates for the spectral Galerkin approximations of the solution of the Navier–Stokes equations (spectral in the sense that one chooses as basis functions the eigenfunctions of the Stokes operator).Error estimates are presented locally in time, valid on a finite interval determined by certain norms of the data. If one assumes the solution to be approximated is uniformly regular fort ≥ 0, the method gives an error estimate which grows exponentially with time. Without further assumptions this is the best estimate that one can expect. However, as pointed out in [2] by J. G. Heywood, if one assumes, additionally, that the solution to be approximated is stable, then one obtains an error estimate which is uniform in time.