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

TL;DR

This paper introduces a variational optimal control approach for inverse Stefan problems, enabling the estimation of heat flux and free boundary without explicit phase transition temperature, and proves convergence of the discretized method.

Contribution

It develops a full discretization scheme for the inverse Stefan problem within an optimal control framework and proves its convergence to the continuous problem.

Findings

Convergence of the discrete optimal control problems to the continuous problem.

Development of an iterative numerical method with low computational cost.

Ability to handle unknown phase transition temperature through measurements.

Abstract

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In {\it Inverse Problems and Imaging, 7, 2(2013), 307-340} we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.