Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element apprbximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domain. Some numerical experiments are carried out at the end of the paper to support our theoretical findings. (C) 2014 IMACS. Published by Elsevier B.V. All tights reserved.