Adaptive Finite Element Methods for Optimal Control

We develop a new approach towards error control and adaptivity for nite element discretizations in optimization problems governed by partial diierential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary opti-mality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. This approach facilitates control of the error with respect to any given quantity of physical interest. The speciic feature introduced by the optimization problem is the natural choice of the error-control functional to conincide with the cost functional of the optimization problem. In this case, the Lagrangian multiplier can directly be used in weighting the cell-residuals in the error estimator. This leads to a particularly simple and cost-eecient algorithm for adapting the mesh according to the particular needs of the optimization problem. This approach is developed and tested for simple model problems in optimal control of semiconductivity.