A priori and a posteriori error analysis of an augmented mixed finite element
method for incompressible fluid flows
Leonardo Figueroa, Gabriel N. Gatica and Norbert Heuer
Comput. Methods Appl. Mech. Engrg. 198 (2), 280-291, 2008.
In this paper we extend recent results on the a priori and a posteriori error
analysis of an augmented mixed finite element method for the linear elasticity
problem, to the case of incompressible materials. Similarly as before, the
present approach is based on the introduction of the Galerkin least-squares
type terms arising from the constitutive and equilibrium equations, and from
the relations defining the pressure in terms of the stress tensor and the
rotation in terms of the displacement, all of them multiplied by stabilization
parameters. We show that these parameters can be suitably chosen so that the
resulting augmented variational formulation is defined by a strongly coercive
bilinear form, whence the associated Galerkin scheme becomes well posed for
any choice of finite element subspaces. Next, we derive a reliable and
efficient residual-based a posteriori error estimator for the augmented mixed
finite element scheme. Finally, several numerical results confirming the
theoretical properties of this estimator, and illustrating the capability of
the corresponding adaptive algorithm to localize the singularities and the
large stress regions of the solution, are also reported.