Finite element approximation of a Stefan problem with degenerate Joule heating

We consider a fully practical finite element approximation of the following degenerate system partial derivative/partial derivativetrho(u)-del.(alpha(u) delu) There Existssigma(u) \delphivertical bar(2), del.(sigma(u) delphi) = 0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, phi. In the above rho(u) is the enthalpy incorporating the latent heat of melting, alpha(u) > 0 is the temperature dependent heat conductivity, and sigma(u) greater than or equal to 0 is the electrical conductivity. The latter is zero in the frozen zone, u less than or equal to 0(1) which gives rise to the degeneracy in this Stefan system. In addition to showing stability bounds, we prove (subsequence) convergence of our finite element approximation in two and three space dimensions. The latter is non-trivial due to the degeneracy in sigma(u) and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.