Stabilization methods of bubble type for the Q(1)/Q(1)-element applied to the incompressible Navier-Stokes equations

In this paper, a general technique is developed to enlarge the velocity spa ce V-h(1) of the unstable Q(1)/Q(1)-element by adding spaces V-h(2) such th at for the extended pair the Babuska-Brezzi condition is satisfied. Example s 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, o ur new elements are much more cheaper. In particular, we shall see that mor e 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 stabil ize the Q1/Q1-element. Moreover, by using the technique of reduced discreti zations and eliminating the additional degrees of freedom we show the relat ionship between enlarging the velocity space and stabilized methods. This r elationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods fo r the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-P itkaranta stabilization and the SUPG method for the incompressible Navier-S tokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized ve rsions of the Navier-Stokes equations but deal with the full nonlinear case . Mathematics Subject Classification. 65N30, 65N12, 76D05.