by Matthies, G., Skrzypacz, P., Tobiska, L..

**Series:** 2003-19, 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