  We consider a pointwise tracking optimal control problem for a semilinear
elliptic partial differential equation. We derive the existence of optimal
solutions and analyze first and, necessary and sufficient, second order
optimality conditions. We devise two strategies of discretization to
approximate a solution of the optimal control problem: a semidiscrete scheme
where the control variable is not discretized -- the so-called variational
discretization approach -- and a fully discrete scheme where the control
variable is discretized with piecewise constant functions. For both solution
techniques, we analyze convergence properties of discretizations and derive
error estimates.
