Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations

In this paper, a gênerai technique is developed to enlarge the velocity space V^ of the unstable Qi/Qi-element by adding spaces V£ such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable éléments which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the AQ1/Q1 -element. However, our new éléments are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Qi to the Q2 and 4Qi, respectively, element are not necessary to stabilize the Q1/Q1 -element. Moreover, by using the technique of reduced discret izat ions and éliminât ing the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular éléments but was not known for quadrilatéral éléments. As a resuit we dérive new stabilized methods for the Stokes and NavierStokes équations. Finally, we show how the Brezzi-Pitkaranta stabilization and the SUPG method for the incompressible Navier-Stokes équations can be recovered as special cases of the gênerai approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes équations but deal wit h the full nonlinear case. Mathematics Subject Classification. 65N30, 65N12, 76D05. Received: November 25, 1998. Revised: July 6, 1999.