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

**Series:** 2007-09, Preprints

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

**Abstract:**

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.

**Keywords:**

Stokes problem, local projection stabilization, equal order interpolation