Thesis Summary:

This thesis considers simplified models that describe oceanic circulation, in particular the linear quasi-geostrophic model in its velocity-pressure and current-vorticity formulations. To address the problems of incomplete initial data, the theory of desensitizing control was used. The main objective was to study the existence of controls that allow to obtain measurements in an observatory, independently of the small variations in the initial conditions. This analysis is reduced to studying problems of approximate or exact controllability. A single continuation property and a reverse inequality were obtained for a Stokes-type cascade system, where the term Coriolis played a determining role. A data assimilation technique was used to determine unknown initial conditions for the purpose of making predictions. This method was implemented numerically. It was possible to determine an approximation of the initial condition from measurements of the state in previous instants of time and knowing the term source, which allowed to make good predictions of the system. In the theoretical aspect, the exact recovery of the initial condition is reduced to a problem of zero controllability, where an inequality of observability was tested. Finally, an adaptive mesh refinement strategy is introduced to reduce oscillations and poor resolution that occurs when the convective term is dominant. An anisotropic indicator of the error a posteriori was proposed to locate the boundary layer without a priori information of the solution and to create meshes well adapted to the solution. This technique was based on the recovery of the hessian solution. Finally, the efficiency of this strategy was determined by means of several numerical experiments.

ISI Publications from the Thesis

Enrique FERNANDEZ-CARA, Galina C. GARCíA, Axel OSSES: Controls insensitizing the observation of a quasi-geostrophic ocean model. SIAM Journal on Control and Optimization, vol. 43, 5, pp. 1616-1639, (2005)

Jose M. CASCON, Galina C. GARCíA, Rodolfo RODRíGUEZ: A priori and a posteriori error analysis for a large-scale ocean circulation finite element model. Computer Methods in Applied Mechanics and Engineering, vol. 192, pp. 5305-5327, (2003)

Enrique FERNANDEZ-CARA, Galina C. GARCíA, Axel OSSES: Insensitizing controls for a large-scale ocean circulation model. Comptes Rendus de l'Académie des Sciences, vol. 337, pp. 265-270, (2003)

No-ISI Publications from the Thesis

Jose M. CASCON, Galina C. GARCíA, Rodolfo RODRíGUEZ: A posteriori error analysis for a large-scale ocean circulation finite element model. Adaptive Modeling and Simulation, pp. 13-14 and CD-ROM. CIMNE, Barcelona, 2003

Other Publications (ISI)

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)