Adaptive finite element methods for optimization problems

We present a new approach to error control and mesh adaptivity in the numerical solution of optimal control problems governed by elliptic differential equations. The indefinite boundary Value problems obtained by the Lagrangian formalism are discretized by the Galerkin finite element method. The mesh adaptation is driven by residual-based a posteriori error estimates which are derived by global duality arguments. This general approach facilitates control of the error with respect to any quantity of physical interest. In discretizing an optimization problem it seems natural to control the error with respect to the given cost functional. In this way, the computed solution can directly be used in weighting the cell residuals in the aposteriori error estimate. This approach has the features of model reduction as used in optimal control of complex dynamical systems. For illustration, we present some results of test computations for simple model problems in optimal control of super-conductivity.