  The numerical simulation of complex physical processes requires the use of
economical discrete models. This lecture presents a general paradigm of
deriving a posteriori error estimates for the Galerkin finite element
approximation of nonlinear problems. Employing duality techniques as used in
optimal control theory the error in the target quantities is estimated in terms
of weighted `primal' and `dual' residuals. On the basis of the resulting local
error indicators economical meshes can be constructed which are tailored to the
particular goal of the computation. The performance of this {\it Dual Weighted
Residual Method} is illustrated for a model situation in computational fluid
mechanics: the computation of the drag of a body in a viscous flow, the drag
minimization by boundary control and the investigation of the optimal
solution's stability.
