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

Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis