Indicadores equilibrados de erro em funcional de interesse para aproximação de problemas elípticos e parabólicos pelo método de Galerkin descontínuo

Abstract

This thesis presents theoretical and practical results on equilibrated error indicators for functional of interest in the approximation of linear elliptic problems and linear and non-linear parabolic problems by the discontinuous Galerkin method. The introduction of the equilibrated fluxes in the representation of the error in the functional of interest allows to improve the quality of the error indicator that is a first important result of this work (in the parabolic case). The second contribution of the work is the approximation of the solution to the dual problem, which enters in the representation of error, by the discontinuous Galerkin method of order higher that of primal method. In this case, the error indicator becomes asymptotically exact. The construction of error indicators for functional of interest for the discontinuous Galerkin method in time and space for linear and nonlinear parabolic problems is the third contribution of this work. All of the above techniques use equilibrated reconstruction of discrete flux in Raviart-Thomas space. Equilibrated flux reconstruction on a specific basis in high-order Raviart-Thomas spaces is another important contribution. Numerical results are presented in the course of the work to demonstrate the efficiency of the developed methods. Error indicators are also used for goal oriented mesh adaptation in various numerical experiments.