The nonlinear multi-dimensional heat transfer problem in discontinuously heterogeneous. but piecewise homogeneous media is treated numerically by using the enthalpy lormulation, certain regularization of the contact conditions between the homogeneous subdomains (like in [1,2]), the fully implicit time discretization and linear finite elements in space with Imear interpolation and numerical integration. A convergence is proved by using a technique that does not check the time derivative of temperature. Phase transitions with a positive latent heat (i.e. Stefan problems) are covered, as well. Besides, the problem need not be of a strongly parabolic type. Some numerical experience with the nonlinear Gauss-Seidel algorithm to solve the created nonlinear algebraic systems is presented, too.