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

¹ Institute of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Praha 1, Czech Republic. (knobloch@karlin.mff.cuni.cz)
² Institute of Analysis and Numerics, Otto von Guericke University, PF 4120, 39016 Magdeburg, Germany. (tobiska@mathematik.uni-magdeburg.de)

Received: 25 November 1998
Revised: 6 July 1999

Abstract

In this paper, a general technique is developed to enlarge the velocity space of the unstable -element by adding spaces such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q₂/Q₁-element and the 4Q₁/Q₁-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 Q₁ to the Q₂ and 4Q₁, respectively, element are not necessary to stabilize the Q₁/Q₁-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.