### 2003-13

#### Finite element error analysis of space averaged flow fields defined by a differential filter

Series: 2003-13, Preprints

MSC:
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
76D05 ~Navier-Stokes equations

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