Numerical methods for Stefan problems with prescribed convection and nonlinear flux

This paper deals with numerical methods for a class of nonlinear degenerate parabolic equations with nonlinear flux boundary conditions, which in particular includes the well-known continuous casting Stefan problem. A practical piecewise linear finite element scheme with numerical integration which is based on an implicit discretization of the diffusion and an explicit approximation of the enthalpy-dependent convection is proposed. The convergence of the scheme is proved without requiring the quasi-uniformity and the acuteness of the finite element meshes. The key ingredient of the analysis is an estimate on the fractional derivative in time of the discrete temperature which provides the necessary strong convergence property. The results of some numerical tests are provided in order to show the accuracy of the scheme.