Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised oseen problem

We consider residual-based stabilised nite element methods for the generalised Oseen problem. The unique solvability based on a modi ed stability condition and an error estimate are proved for inf-sup stable discretisations of velocity and pressure. The analysis highlights the role of an additional stabilisation of the incompressibility constraint. It turns out that the stabilisation terms of the streamline-di usion (SUPG) type play a less important role. In particular, there exists a conditional stability problem of the SUPG stabilisation if two relevant problem parameters tend to zero. The analysis extends a recent result to general shape-regular meshes and to discontinuous pressure interpolation. Some numerical observations support the theoretical results.