Finite Element Error Estimates for Optimal Control Problems with Pointwise Tracking

We consider a linear-quadratic elliptic optimal control problem with point evaluations of the state variable in the cost functional. The state variable is discretized by conforming linear finite elements. For control discretization, three different approaches are considered. The main goal of the paper is to significantly improve known a priori discretization error estimates for this problem. We prove optimal error estimates for cellwise constant control discretizations in two and three space dimensions. Further, in two space dimensions, optimal error estimates for variational discretization and for the post-processing approach are derived.