TABLE OF CONTENTS In the study of convergence for saddle point problems we assumed that the finite element spaces for velocities and pressure satisfy the inf-sup condition. This raises the question of whether this condition is only needed to get a complete mathematical theory, or whether it plays an essential role in practice.
The answer to this question is given by a well-known finite element method for which the Brezzi condition is violated. Although instabilities had been observed in computations with this element in fluid mechanics, attempts to explain its instable behavior and to overcome it in a simple way mostly proved to be unsatisfactory. The Brezzi condition turned out to be the appropriate mathematical tool for understanding and removing this instability, and it also provided the essential breakthrough in practice. There are very few areas [7] where the mathematical theory is of as great importance for the development of algorithms as in fluid mechanics.
After discussing the instable element mentioned above, we present two commonly used stable elements and another one which is easier to implement. There is also a nonconforming divergence-free element which allows the elimination of the pressure.
In the Stokes equation (6.1), ? u and grad p are the terms with derivatives of highest order for the velocity and pressure, respectively. Thus, the orders of the differential operators differ by 1. This suggests the rule of thumb: the degree of the polynomials used to approximate...
