Convergence Of Goal Oriented Adaptive Finite Element Methods

In this thesis we discuss convergence theory for goal-oriented adaptive finite element methods for second order elliptic problems. We develop results for both linear nonsymmetric and semilinear problems. We start with a brief description of the finite element method applied to these problems and some basic error estimates. We then provide a detailed error analysis of the method as described for each problem. In each case, we establish convergence in the sense of the quantity of interest with a goal-oriented variation of the standard adaptive finite element method using residual-based indicators. In the linear case we establish the adjoint as the appropriate differential operator for the dual problem. We establish contraction of the quasi-error for each of the primal and dual problems yielding convergence in the quantity of interest. We follow these results with a complexity analysis of the method. In the semilinear case we introduce three types of linearized dual problems used to establish our results. We give a brief summary of a priori estimates for this class of problems. After establishing contraction results for the primal problem, we then provide additional estimates to show contraction of the combined primal and dual system, yielding convergence of the goal function. We support these results with some numerical experiments. Finally, we include an appendix outlining some common methods used in a posteriori error estimation and briefly describe iterative methods for solving nonlinear problems.