A Unified Convergence Analysis for Local Projection Stabilisations Applied to the Oseen Problem

The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach. Mathematics Subject Classification. 65N12, 65N30, 76D05. Received October 6, 2006. Revised February 20, 2007. Introduction The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur in case of higher Reynolds numbers due to the dominating convection. The idea of streamline upwind Petrov-Galerkin (SUPG) stabilisation has been proposed for the advective term in [12] and extended to the Stokes equations in [27] where a pressure stabilisation Petrov-Galerkin (PSPG) method is considered accommodating low equal-order interpolation to be stable and convergent. This formulation circumvented the need to satisfy the discrete inf-sup condition for many interpolations. In an attempt to get the stability features of these works, a method is proposed in [16] that is at the same time advective stable and overcomes the inf-sup