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Local projection stabilization

by Ganesan, S., Matthies, G., Tobiska, L..

Series: 2007-09, Preprints

65N12 Stability and convergence of numerical methods
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

The local projection stabilization allows to circumvent the
Babu\v{s}--Brezzi condition and to use equal order interpolation
for discretizing the Stokes problem. The projection is usually done
in a two-level approach by projecting the pressure gradient onto a
discontinuous finite element space living on a patch of elements.
We propose a new local projection stabilization method based on (possibly)
enriched finite element spaces and discontinuous projection spaces defined on
the same mesh. Optimal order of convergence is shown for pairs of
approximation and projection spaces satisfying a certain inf-sup condition.
Examples are enriched simplicial finite elements and standard
quadrilateral/hexahedral elements. The new approach overcomes
the problem of an increasing discretization stencil and, thus, is
simple to implement in existing computer codes. Numerical tests confirm the
theoretical convergence results which are robust with
respect to the user-chosen stabilization parameter.

Stokes problem, local projection stabilization, equal order interpolation