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Local projection type stabilisation applied to inf-sup stable

by Matthies, G., Tobiska, L..

Series: 2007-47, Preprints

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

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.

Stabilised finite elements, Navier--Stokes equations,