Adaptive Algorithms for Goal-Oriented Error Estimation: Comparison of Different Error Representations

Abstract

This thesis focuses on approximating a quantity of interest Q(u), where u is the solution to a Partial Differential Equation (PDE). With this goal, it compares various approaches to adaptive algorithms based on the Finite Element (FE) method. Specifically, it examines several globally equivalent error representations used in goal-oriented adaptivity. It borrows previously developed error representations and draws a comparison of their efficiency and accuracy with newly introduced representations. The error representations are first defined in the context of an abstract problem using duality techniques and Galerkin orthogonality properties. They are then explicitly defined in the context of two problems: a general 1D convection-diffusion-reaction problem, and a 2D Poisson problem. These representations are incorporated into h-adaptive algorithms for a 1D boundary-layer problem and a 2D Poisson problem, as well as a p-adaptive algorithm for the 1D Helmholtz problem. The adaptive algorithms are tested with different marking strategies and for different quantities of interest. Numerical results highlight the local variation of the error representations and demonstrate their influence on the performance of the algorithms. Finally, these results are used to classify the error representations according to their efficiency and accuracy.