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2006-44

A unified convergence analysis for local projection stabilisations

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