This thesis deals with the numerical solution of optimization problems in function spaces governed by linear elliptic partial differential equations. Many physical processes for instance in thermodynamics, elasticity, fluid mechanics or electrical engineering are modeled by partial differential equations. The aim of optimal control is to regulate occurring parameters or other quantities in such a way that the result of the mathematical model is optimal in a certain sense. In particular, Neumann boundary control problems are investigated meaning that the flux of the state variable on the boundary of the underlying computational domain can be controlled. Of particular interest are finite element discretizations for these problems on domains having polyhedral shape. Since singularities in a vicinity of corners and edges are expected to be contained in the solution, optimal convergence of the finite element method on quasi-uniform meshes can as a general rule not be guaranteed. For a better description of the occurring singularities weighted Sobolev spaces are used in this thesis, and, exploiting corresponding regularity results finite element error estimates are proved. Up to now sharp error estimates for the trace of the finite element approximation of the Neumann problem on polyhedral domains were unknown, and the newly developed estimates in this thesis allow an improvement of many convergence results for Neumann boundary control problems. Among others problems with $L^2(\Gamma)$-regularization are considered and improved estimates for the numerical approximation using the full discretization, the postprocessing approach and the variational discretization are derived. Further, a new energy regularization approach is considered on polygonal domains, where the convergence rate depends in this case solely on the interior angles at the corners of the domain. The aim of this thesis is always to investigate which convergence rate can be expected on quasi-uniform meshes, and to what extend the best-possible convergence rate can be retained with local mesh refinement, such that methods for the numerical computation of Neumann boundary control problems can be improved significantly.     «

This thesis deals with the numerical solution of optimization problems in function spaces governed by linear elliptic partial differential equations. Many physical processes for instance in thermodynamics, elasticity, fluid mechanics or electrical engineering are modeled by partial differential equations. The aim of optimal control is to regulate occurring parameters or other quantities in such a way that the result of the mathematical model is optimal in a certain sense. In particular, Neumann...     »