Thesis Summary:

The present work consists of two parts clearly defined. In the first part we use a mixed formulation to analyze the numerical resolution of certain class of nonlinear elliptic boundary value problems, on a Lipschitz domain in the plane. More precisely, we consider a nonlinear exterior transmission problem with discontinuities, whose discrete variational formulation is obtained by coupling the mixed finite element method with the boundary integral method. We show that the discrete scheme is well-posed and prove optimal rates of convergence. In addition, we present an a-posteriori error analysis for this formulation, subject that had not been developed yet for transmission problems. Several numerical examples confirm our theoretical results and provide empiric evidences of an eventual efficiency of the a-posteriori error estimator. In the second part of this thesis, we extend the applicability of the local discontinuous Galerkin method to nonlinear boundary value problems. First, we consider a nonlinear diffusion problem with mixed boundary conditions, and then we study a class of quasi-newtonian Stokes fluids in stationary regime. We prove that the discrete schemes are well-posed and provide the corresponding a-priori error estimates. In addition, we also develop a-posteriori error analyses yielding reliable estimators for both problems, and present several numerical results illustrating the performance of the a-posteriori error estimators.

ISI Publications from the Thesis

Rommel BUSTINZA, Gabriel N. GATICA: A mixed local discontinuous Galerkin method for a class of nonlinear problems in fluid mechanics. Journal of Computational Physics, vol. 207, 2, pp. 427-456, (2005)

Rommel BUSTINZA, Gabriel N. GATICA: A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM Journal on Scientific Computing, vol. 26, 1, pp. 152-177, (2004)

Rommel BUSTINZA, Galina C. GARCíA, Gabriel N. GATICA: A mixed finite element method with Lagrange multipliers for nonlinear exterior transmission problems. Numerische Mathematik, vol. 96, 3, pp. 481-523, (2004)

No-ISI Publications from the Thesis

Rommel BUSTINZA, Bernardo COCKBURN, Gabriel N. GATICA: An a-posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. Journal of Scientific Computing, vol. 22, 1, pp. 147-185, (2005)

Other Publications (ISI)

Rommel BUSTINZA, Gabriel N. GATICA, María GONZáLEZ: A mixed finite element method for the generalized Stokes problem. International Journal for Numerical Methods in Fluids, vol. 49, 8, pp. 877-903, (2005)

Rommel BUSTINZA, Gabriel N. GATICA, María GONZáLEZ, Salim MEDDAHI, Ernst P. STEPHAN: Enriched finite element subspaces for dual-dual mixed formulations in fluid mechanics and elasticity. Computer Methods in Applied Mechanics and Engineering, vol. 194, 2-5, pp. 427-439, (2005)