THE POSTPROCESSED MIXED FINITE-ELEMENT METHOD FOR THE NAVIER–STOKES EQUATIONS: REFINED ERROR BOUNDS

摘要

A postprocessing technique for mixed finite-element methods for the incompressible Navier–Stokes equations is analyzed. The postprocess, which amounts to solving a (linear) Stokes problem, is shown to increase the order of convergence of the method to which it is applied by one unit (times the logarithm of the mesh diameter). In proving the error bounds, some superconvergence results are also obtained. Contrary to previous analysis of the postprocessing technique, in the present paper we take into account the loss of regularity suffered by the solutions of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. As in [H. G. Heywood and R. Rannacher, SIAM J. Numer. Anal., 25 (1988), pp. 489–512], where the same hypothesis is assumed, no better than fifth-order convergence is achieved.