Error estimates for finite element approximation of optimal control problems

Dr. Daniel Wachsmuth

May 5, 2009, 1:30 p.m. HF 136

We consider the numerical approximation of simple optimal control problems: a quadratic functional is to be minimized subject to a linear elliptic equation and pointwise inequality constraints on control and state. This problem is then discretized by means of finite-elements. A particular discretization scheme - the variational control discretization of M.Hinze - is introduced. A-priori error estimates are proven with the help of first-order necessary optimality conditions. Moreover, goal-oriented a-posteriori error estimates are discussed.

The presented methods rely on the convexity of the underlying optimization problem. However, for non-linear equations or non-quadratic objective functionals this optimization problem is not convex in general. We will discuss the arising difficulties at the end of the talk.