Numerical methods for a class of nonlinear degenerated parabolic equations with nonlinear flux boundary conditions are considered. In particular, these equations include the well-known continuous casting Stefan problem. A piecewise linear finite element scheme with numerical integration based on an implicit discretization of the diffusion and explicit approximation of the enthalpy dependent convection is developed.
The convergence of the finite element scheme is proved. The key ingredient of the analysis is an estimate of the fractional derivative in time of the discrete temperature which provides the necessary strong convergence property. Some results of numerical tests are presented which show the accuracy of the scheme.