by Matthies, G., Skrzypacz, P., Tobiska, L..

**Series:** 2006-44, Preprints

- MSC:
- 65N12 Stability and convergence of numerical methods
- 65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 76D05 ~Navier-Stokes equations

**Abstract:**

The discretisation of the Oseen problem by finite element methods suffers

in general from two reasons. First, the discrete inf-sup (Babu\v{s}ka--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 modeling introduced by

Guermond is shown. This clarifies the relation of the

local projection stabilisation to the variational multiscale

approach.

**Keywords:**

Stabilised finite elements, Navier--Stokes equations, equal-order