  We consider an elliptic optimal control problem where the objective
functional contains evaluations of the state at a finite number of points. In
particular, we use a fidelity term that encourages the state to take certain
values at these points, which means our problem is related to ones with state
constraints at points. The analysis and numerical analysis differs from when
the fidelity is in the $L^2$ norm because we need the state space to embed into
the space of continuous functions. In this paper we discretise the problem
using two different piecewise linear finite element methods. For each
discretisation we use two different approaches to prove a priori $L^2$ error
estimates for the control. We discuss the differences between these methods and
approaches and present numerical results that agree with our analytical
results.
