Smoothing and Rotheʹs method for Stefan problems in enthalpy form

The classical two-phase Stefan problem as well as its weak variational formulation model the connection between the different phases of the considered material by interface conditions at the occurring free boundary or by a jump of the enthalpy. One way to treat the corresponding discontinuous variational problems consists in its embedding into a family of continuous ones and applying some standard techniques to the chosen approximation problems. The aim of the present paper is to analyze a semi-discretization via Rotheʹs method and its convergence behavior in dependence of the smoothing parameter. While in Grossmann et al. (Optimization, in preparation) the treatment of the Stefan problem is based on the given variable, i.e. the temperature, here first a transformation via the smoothed enthalpy is applied. Numerical experiments indicate a higher stability of the discretization by Rotheʹs method. In addition, to avoid inner iterations a frozen coefficient approach as common in literature is used.