### 2007-47

by Matthies, G., Tobiska, L..

**Series:** 2007-47, Preprints

- MSC:
- 65N12 Stability and convergence of numerical methods
- 65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 65N15 Error bounds

**Abstract:**

The local projection method is applied to inf-sup stable discretisations of

the Oseen problem. Error bounds of order $r$ are proven for known inf-sup

stable pairs of finite element spaces which approximate velocity and

pressure by elements of order $r$ and $r-1$, respectively.

In case of a positive reaction coefficient, the error constants are robust

with respect to the viscosity but depend on the positive lower bound

of the reaction coefficient. Using enriched velocity spaces, error estimates

of order $r$ are established which are also robust when both the viscosity

and the reaction coefficient tend to zero. Moreover, for certain

velocity and pressure approximations by elements of order $r$,

the discrete inf-sup condition holds and a robust error estimate of improved

order $r+1/2$ is shown.

Numerical results confirm the theoretical convergence results.

**Keywords:**

Stabilised finite elements, Navier--Stokes equations,