The authors deal with a discontinuous parabolic problem which arises from the enthalpy formulation of Stefan problems. They apply a technique of penalty methods which forms a smooth approximation of an exact penalty to construct the embedded problems. As another significant problem in smoothing techniques they study the convergence behaviour of Newton’s method applied to the nonlinear minimization at different time levels. The subproblems at each time level can be considered as nonsmooth convex variational problems of obstacle type.