  We present a finite element method along with its analysis for the optimal
control of a model free boundary problem with surface tension effects,
formulated and studied in \cite{HAntil_RHNochetto_PSodre_2014a}. The state
system couples the Laplace equation in the bulk with the Young-Laplace equation
on the free boundary to account for surface tension. We first prove that the
state and adjoint system have the requisite regularity for the error analysis
(strong solutions). We discretize the state, adjoint and control variables via
piecewise linear finite elements and show optimal $O(h)$ error estimates for
all variables, including the control. This entails using the second order
sufficient optimality conditions of \cite{HAntil_RHNochetto_PSodre_2014a}, and
the first order necessary optimality conditions for both the continuous and
discrete systems. We conclude with two numerical examples which examine the
various error estimates.
