Numerical Analysis of a Stefan Problem

We can observe many phenomena involving Free Boundaries in various fields of engineering and applied sciences, for example, free boundary problems in optimum design, the pollution of air and water, the equilibrium of plasma. For such problems it is important to develop reliable computational methods which are practically efficient in applications. Naturally, a crucial point in numerical methods for free boundary problem is how to deal with the moving boundary. In order to meet this difficulty, various approaches have been proposed they can be classified into two groups. One of them is to follow the free boundary directly. The other is to transform the original problem into an auxiliary problem with a fixed boundary. The authors technique uses the penalty method and falls into the second group. In this paper they apply this method to the Stefan problem which arises in the analysis of melting of ice adjacent to a heated body of water. The essential idea of their method is to transform the Stefan problem into an initial-boundary value problem for the heat equation defined in a cylindrical domain, occupied jointly by water and ice, with an artificial heat absorption in the ice region. Their formulation is closely related to the approach using variational inequalities.