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Stabilized Methods of Bubble Type for the Q1/Q1-Element Applied to the Incompressible Navier-Stokes Equations

by Knobloch, Petr, Tobiska, Lutz.

Series: 1998-35, Preprints

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

In this paper, a general technique is developed to enlarge the velocity space
$\Vhj$ of the unstable $Q_1/Q_1$--element by adding spaces $\Vhd$ such that
for the extended pair the Babu\v{s}ka--Brezzi condition is satisfied. Examples
of stable elements which can be derived in such a way imply the stability of
the well--known
$Q_2/Q_1$--element and the $4Q_1/Q_1$--element. However, our new elements
are much more cheaper. In particular, we shall see that more than half of the
additional degrees of freedom when switching from the $Q_1$ to the $Q_2$ and
$4Q_1$, respectively, element are not necessary to stabilize the
$Q_1/Q_1$--element. Moreover, by using the technique of reduced discretizations
and eliminating the additional degrees of freedom we show
the relationship between enlarging the velocity space and stabilized methods.
This relationship has been established for triangular elements but was not
known for quadrilateral elements. As a result we derive new stabilized
methods for the Stokes and Navier--Stokes equations. Finally, we show
how the Brezzi--Pitkäranta stabilization and the SUPG method for the
incompressible Navier--Stokes equations can be recovered as special cases of
the general approach. In contrast to earlier papers we do not restrict
ourselves to linearized versions of the Navier--Stokes equations but deal
with the full nonlinear case.

Babu\v{s}ka--Brezzi condition, stabilization, Stokes equations, Navier--Stokes equations