by Knobloch, Petr, Tobiska, Lutz.

**Series:** 1998-35, Preprints

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

**Abstract:**

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.

**Keywords:**

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