Preprint series: 03-19 , Preprints

MSC:

65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

65N12 Stability and convergence of numerical methods

65N15 Error bounds

Abstract: For the Poisson equation on uniform meshes it is well-known that the
piecewise linear conforming finite element solution approximates the
interpolant to a higher order than the solution itself. In this paper,
this type of superclose property is established for a nonstandard
interpolant of the $Q_2-P^{\textrm{disc}}_1$ element applied to the stationary
Stokes and Navier--Stokes problem, respectively. Moreover, applying a
$Q_3-P^{\textrm{disc}}_2$ post-processing technique, we can also state a
superconvergence property for the discretisation error of the post-processed
discrete solution to the solution itself. Finally, we show that inhomogeneous
boundary values can be approximated by the standard Lagrange
$Q_2$-interpolation without influencing the superconvergence property.
Numerical experiments verify the predicted convergence rates.

Keywords: finite elements, Navier--Stokes equations, superconvergence, postprocessing

Upload: 2003-07-01-07-01