The supercloseness property of the Stoke projection for the transient Navier-Stokes equations and global superconvergence analysis

In this paper, we derive the supercloseness properties and global superconvergence results for the implicit Euler scheme of the transient Navier-Stokes equations. Using a prior estimate of finite element solutions, the properties of the Stokes projection and Stokes operator, the derivative transforming skill and the H -1 -norm estimate, we deduce the supercloseness properties of the Stokes projection for the velocity in L∞ (H 1 )-norm and pressure in L∞ (L 2 )-norm. Then the supercloseness properties of the interpolation operators are obtained for two pairs of retangular element: the bilinear-constant element and the Bernadi-Raugel element. Finally, by the interpolation postprocessing technique, we obtain the global superconvergent results. The supercloseness analysis is based on the Stokes projection, which makes the proof more concise. Compared with previous superconvergence results, a lower regularity of solution is needed, and no time step restrict is required.