Local projection type stabilisation applied to inf-sup stable discretisations of the Oseen problem

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.