## 1 Introduction
As already mentioned while discussing the interface tracking methods, Stefan problem can be treated by using standard FEM and a mesh which conforms to the phase interface (e.g. [14]). The heat-conduction problem is then solved separately in the solid and liquid domains by imposing essential boundary conditions on the phase-interface, i.e. by prescribing the phase-transition temperature. Then, the evolution of the interface is obtained from the jump in the heat flow across the interface, and the position of the nodes belonging to the interface is updated accordingly. The jump in the temperature gradient across the phase interface is possible thanks to the adapted mesh but, of course, remeshing is required every few discrete time-steps. Moreover, each time a new mesh is generated, the nodal values of the temperature must be recomputed by using the interpolated values from the older mesh. In order to avoid remeshing it is possible to enhance the FEM by allowing for discontinuities that cross the elements. This can be achieved either by embedding the discontinuity directly within the finite elements intersected by the phase-interface and controlling it by additional _elemental_ degrees of freedom, or by enriching the interpolation by means of nodal extended shape functions controlled by additional _nodal_ degrees of freedom. The former approach we call _embedded-discontinuity FEM_, while the latter is well known as _eXtended Finite Element Method_ (XFEM). In this paper, we focus on these two methods by reviewing the XFEM and adapting it to a specific case, and by developing an original embedded-discontinuity FEM. In both methods, the nodes of the discretisation are fixed and no remeshing is required. It is then natural to adopt an Eulerian approach for the interface description, so that the level set method is used. The XFEM has been applied to Stefan problem by [PERSON] and [PERSON] (2002) [15], who focus on the one-dimensional case, by [PERSON] et al. (2002) [16], who develop the general method that is adapted here for the use with different constraint enforcement methods, and by [PERSON] et al. (2006) [17], who introduce a simplified case of fluid motion and dendritic solidification by using a smeared-interface model. An application of XFEM to the problem of biofilm growth in two dimensions is developed in Reference [18]. No applications of the embedded-discontinuity FEM to Stefan problem are known to the writers. Though, as it has been shown, the research is currently tackling planar and spatial cases, the one-dimensional Stefan problem is the subject of several recent research works [11, 12, 15] and of a monographic book [19]. Also the examples that are presented in this paper are developed for cases that can be treated as one dimensional due to planar, cylindrical, or spherical symmetry. This is done for the sake of simplicity in exposition, implementation, and result visualisation, which allow concentrating on the differences between different approaches of the XFEM and between XFEM and embedded-discontinuity FEM. Moreover, exact analytical solutions are available in some one-dimensional cases, so that error and convergence can be studied. In further cases, it is possible to develop, as it is done in Sections 6 and 7, simple approximated analytical solutions in the so-called pseudo steady state approximation, in which the heat-conduction problem and the evolution of the interface can be decoupled. Though the aim of the present one-dimensional analyses is basically heuristic, they may also find practical applications. Examples in planar, cylindrical, and spherical symmetry are, respectively, the maintenance of ice surface of skating facilities, the freezing of water into pipelines, or the solidification of droplets of liquid metal into cooling fluid. The initial review provided in this paper includes details that are not strictly necessary for the subsequent one-dimensional applications but that are advisable for a clearer understanding and propaedeutical for the discussion. This paper is organised as follows.