03-19
Superconvergence of a 3d finite element method for stationary Stokes and Navier--Stokes problems
by Matthies, G.; Skrzypacz, P.; Tobiska, L.
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
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