# Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)

## Abstract

Adaptive finite element methods for the numerical solution of partial differential equations consist of successive cycles of the loop
$${\rm SOLVE}\ \Longrightarrow\ {\rm ESTIMATE}\ \Longrightarrow\ {\rm MARK}\ \Longrightarrow\ {\rm REFINE}.$$

Here, SOLVE stands for the finite element solution of the problem with respect to a given triangulation of the computational domain. The following step ESTIMATE is devoted to the estimation of the global discretization error in some appropriate norm or a user specified quantity of interest by a cheaply computable a posteriori error estimator. The estimator is assumed to consist of local contributions whose actual magnitude is then used in the step MARK to specify elements of the triangulation for refinement. The final step REFINE deals with the generation of a new triangulation based on the refinement of the elements selected in the previous step according to specific refinement rules. Adaptive finite elements are by now well established. There are various approaches such as residual-type a posteriori error estimators which rely on the proper evaluation of the residuals with respect to a computed approximation in the norm of the dual space and hierarchical type estimators where the equation satisfied by the error is suitably localized along with a solution of the local problems by higher order finite elements (cf., e.g. [1, 3, 35]). Averaging-type estimators typically use some sort of gradient recovery on element-related patches (cf., e.g. [1, 35]), whereas the theory of guaranteed error majorants provides reliable upper bounds for the error (see ). Finally, the goal oriented weighted dual approach extracts information on the error via the dual problem (cf. [4, 12]).

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