Error analysis of some Galerkin least squares methods for the elasticity equations

The recent technique of stabilizing mixed finite element methods by augmenting the Galerkin formulation with least squares terms calculated separately on each element is considered. The error analysis is performed in a unified manner yielding improved results for some methods introduced earlier. In addition, a new formulation is introduced and analyzed. method is used within two different contexts. The first is in connection with the plain strain or three-dimensional problems near the incompressible limit for which the Poisson ratio equals one half. It is well known that for these problems a standard lower-order displacement finite element method will break down. By now, it is also well known how a working method can be obtained; a new unknown, the "pressure," is introduced and discretized with a finite element space different from that used for the displacements. The finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by Brezzi and Babuska (5), (7). The application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition. For a particular choice of displacement and pressure approximations it can be a difficult task to verify this "Babuska-Brezzi" or "inf-sup" condition. During the last decade this problem has been intensively studied and the performances of many combinations are now known; cf. (18) and (23), (26) for some recent extensions. In addition there exists a general technique, utilizing "bubble functions," by which a stable method can