Thesis Summary:

In this thesis we build on recent results on the a priori and a posteriori error analysis of a so called augmented mixed finite element method for the linear elasticity problem in three distinct directions. The first of those directions is the derivation of such an augmented mixed finite element method for the incompressible linear elasticity problem and the corresponding a priori error analysis. The incompressible linear elasticity problem can be seen as a limiting case of the standard linear elasticity problem, yet crucially, the incompressibility condition to be fulfilled demands a treatment of its own. 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 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. The second direction, closely related to the first, is the derivation of a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme for the incompressible linear elasticity problem. An adaptive algorithm is then proposed and shown to be capable of localizing singularities and large stress regions of the solution. Finally, for the third direction, we focus on the augmented mixed finite element method for the standard linear elasticity problem and propose the use of the preconditioned GMRES method to solve efficiently the large and sparse linear systems that arise. The spectral properties of the stiffness matrix are used to show how standard preconditioners can directly be used.Numerical examples are provided for each followed direction.

ISI Publications from the Thesis

Leonardo E. FIGUEROA, Gabriel N. GATICA, Antonio MARQUEZ: Augmented mixed finite element methods for the stationary Stokes equations. SIAM Journal on Scientific Computing, vol. 31, 2, pp. 1082-1119, (2008).

Leonardo E. FIGUEROA, Gabriel N. GATICA, Norbert HEUER: A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 2, pp. 280-291, (2008).