Adaptive finite element methods for optimal control of partial differential equations: Basic concept

A new approach to error control and mesh adaptivity is described for the di scretization of optimal control problems governed by elliptic partial diffe rential equations. The Lagrangian formalism yields the first-order necessar y optimality condition in form of an indefinite boundary value problem whic h is approximated by an adaptive Galerkin finite element method. The mesh d esign in the resulting reduced models is controlled by residual-based a pos teriori error estimates. These are derived by duality arguments employing t he cost functional of the optimization problem for controlling the discreti zation error. In this case, the computed state and costate variables can be used as sensitivity factors multiplying the local cell-residuals in the er ror estimators. This results in a generic and simple algorithm for mesh ada ptation within the optimization process. This method is developed and teste d for simple boundary control problems in semiconductor models.