03-13
Finite element error analysis of space averaged flow fields defined by a differential filter
Preprint series: 03-13, Preprints
- MSC:
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 76D05 Navier-Stokes equations, See also {35Q30}
Abstract: This paper analyses finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier-Stokes equations with a differential filter. It is shown that $\|\overline{\bu} -\overline{\bu^h}\|_{L^2}$, the error of the filtered velocity $\overline{\bu}$ and the filtered finite element approximation of the velocity $\overline{\bu^h}$, converges under certain conditions of higher order than $\|{\bu} -{\bu^h}\|_{L^2}$, the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the $L^2$-error of finite element approximations of $\overline{\bu}$ and $\overline{\bu^h}$ is considered. Numerical tests in two and three space dimensions support the analytical results.
Keywords: differential filter, convergence of finite element method
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