### 2003-13

by Dunca, A., John, V..

**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