Abstract Within the framework of the finite element method, we present in this paper an efficient algorithm for solving nonlinear heat problems involving phase change. In particular, two problems are considered in this work: the stationary convection–diffusion problem, and the classical transient heat problem. The mathematical models used to solve these problems are based upon enthalpy formulations. The algorithmic design is based on a Newton-type iterative procedure for the stationary problem, and for the transient one, a combination with classical finite difference schemes in time is performed. The proposed phase change algorithmic treatment is applicable for both the situations in which the latent heat takes place over a temperature range or at fixed temperature. Hence, for this latter situation, a regularization over a narrow temperature range is not necessary. The numerical implementation is discussed in detail where we include also the other possible nonlinearities, namely, the temperature dependence of the thermal conductivity and the radiation-type boundary conditions. Finally, a set of representative numerical simulations to illustrate the effectiveness of the proposed method is given.