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2003-19

Superconvergence of a 3d finite

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