# Superconvergence of a 3D finite element method for stationary Stokes and Navier-Stokes problems

G. Matthies, P. Skrzypacz, L. Tobiska

Research output: Contribution to journalArticle

8 Citations (Scopus)

### Abstract

For the Poisson equation on rectangular and brick 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 article, this type of supercloseness property is established for a special interpolant of the Q2 - P1 disc element applied to the 3D stationary Stokes and Navier-Stokes problem, respectively. Moreover, applying a Q3 - P2 disc postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q 2-interpolation without influencing the super-convergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost-benefit analysis between the two third-order methods, the post-processed Q2 - P1 disc discretization, and the Q 3 - P2 disc discretization is carried out.

Original language English 701-725 25 Numerical Methods for Partial Differential Equations 21 4 https://doi.org/10.1002/num.20058 Published - Jul 2005 Yes

### Fingerprint

Navier-Stokes Problem
Superconvergence
Interpolants
Stokes
Discretization
Finite Element Method
Third-order Method
Cost-benefit Analysis
Finite element method
Discretization Error
Finite Element Solution
Poisson's equation
Boundary Value
Post-processing
Lagrange
Piecewise Linear
Convergence Properties
Convergence Rate
Interpolate
Numerical Experiment

### Keywords

• Finite elements
• Navier-Stokes equations
• Postprocessing
• Superconvergence

### ASJC Scopus subject areas

• Analysis
• Applied Mathematics
• Computational Mathematics

### Cite this

In: Numerical Methods for Partial Differential Equations, Vol. 21, No. 4, 07.2005, p. 701-725.

Research output: Contribution to journalArticle

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