The supercloseness property of the Stoke projection for the transient
Navier-Stokes equations and global superconvergence analysis
Abstract
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∞(H1)-norm and pressure in
L∞(L2)-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.