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We develop a new variational formulation of the inverse Stefan problem, whereinformation on the heat flux on the fixed boundary is missing and must be foundalong with the temperature and free boundary. We employ optimal controlframework, where boundary heat flux and free boundary are components of thecontrol vector, and optimality criteria consist of the minimization of the sumof $L_2$-norm declinations from the available measurement of the temperatureflux on the fixed boundary and available information on the phase transitiontemperature on the free boundary. This approach allows one to tackle situationswhen the phase transition temperature is not known explicitly, and is availablethrough measurement with possible error. It also allows for the development ofiterative numerical methods of least computational cost due to the fact thatfor every given control vector, the parabolic PDE is solved in a fixed regioninstead of full free boundary problem. In {\it Inverse Problems and Imaging, 7,2(2013), 307-340} we proved well-posedness in Sobolev spaces framework andconvergence of time-discretized optimal control problems. In this paper weperform full discretization and prove convergence of the discrete optimalcontrol problems to the original problem both with respect to cost functionaland control.

On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences