Thesis Summary:

In this thesis we develop the analysis of a-priori and a-posteriori error of some stabilized mixed finite element methods. For this purpose we consider the following model problems: A Poisson problem with mixed boundary conditions. A Poisson problem with Neumann type conditions. A linear elasticity problem with Dirichlet type conditions. For the first problem we present a new mixed formulation augmented with Lagrange multiplier that allows us to analyze its numerical resolution. Specifically, the augmented scheme is deduced by introducing least squares residual terms from the constitutive and equilibrium equations. We use the classical Babuvska-Brezzi theory to demonstrate that the resulting dual mixed formulation and its corresponding Galerkin scheme are well-posed problems, and we provide the optimal convergence ratios. Then, we develop the a-posteriori error analysis of two different estimators, one of residual type, which proves to be reliable and efficient, and another estimator based on the error Ritz projection, which proves to be reliable and quasi-efficient. Finally, we include numerical results that support the efficiency of both adaptive schemes. For the second problem we present the a-priori and a-posteriori error analysis of a new stabilized scheme, which introduces the boundary solution as a Lagrange multiplier. This suggests us to enrich the formulation with a residual term measured in Sobolev space norm of order 1/2. We used baselines to construct a bilinear form, equivalent to the respective scalar product, that allows controlling this term stabilizer. We prove that both the variational formulation and the associated Galerkin scheme are well-proposed problems, and we deduce the corresponding optimum convergence ratios. In addition, we present the analysis of an a-posteriori error estimator that proves to be reliable and quasi-efficient. Finally, for the elasticity problem, we consider a new augmented formulation that arises by including least squares terms from the constitutive and equilibrium equations and from the relation that defines the rotation in terms of the displacements. For this formulation we developed a reliable and efficient residual type error estimator. We present numerical results that confirm the theoretical properties of the estimator and the versatility of the adaptive scheme.

ISI Publications from the Thesis

Tomás BARRIOS, Gabriel N. GATICA, Freddy PAIVA: A-priori and a-posteriori error analysis of a wavelet-based stabilization for the mixed finite element method. Numerical Functional Analysis and Optimization, vol. 28, 3-4, pp. 265-286, (2007)

Tomás BARRIOS, Gabriel N. GATICA: An augmented mixed finite element method with Lagrange multipliers: a-priori and a-posteriori error analyses. Journal of Computational and Applied Mathematics, vol. 200, 2, pp. 653-676, (2007)

Tomás BARRIOS, Gabriel N. GATICA, María GONZáLEZ, Norbert HEUER: A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity. Mathematical Modelling and Numerical Analysis, vol. 40, 5, pp. 843-869, (2006)

Tomás BARRIOS, Gabriel N. GATICA, Freddy PAIVA: A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers. Applied Mathematics Letters, vol. 19, 3, pp. 244-250, (2006)

Other Publications (ISI)

Tomás BARRIOS, Gabriel N. GATICA, Luis F. GATICA: On the numerical analysis of a nonlinear elliptic problem via mixed-FEM and Lagrange multipliers. Applied Numerical Mathematics, vol. 48, 2, pp. 135-155, (2004)

Rodolfo ARAYA, Tomás BARRIOS, Gabriel N. GATICA, Norbert HEUER: A-posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem. Computer Methods in Applied Mechanics and Engineering, vol. 191, 21-22, pp. 2317-2336, (2002)