We consider a fully practical finite element approximation of the
following degenerate system
$$
{\frac{\partial }{\partial t}} \rho(u)
- \nabla . ( \,\alpha(u) \,\nabla u ) \ni \sigma(u)\,|\nabla\phi|^2 ,
\quad \nabla . (\, \sigma(u) \,\nabla \phi ) = 0
$$
subject to an initial condition on the temperature, u,
and boundary conditions on both u
and the electric potential, ϕ.
In the above
p(u) is the enthalpy
incorporating the latent heat of melting, α(u) > 0 is
the temperature dependent heat conductivity, and σ(u) > 0
is the electrical
conductivity. The latter is zero in the frozen zone, u ≤ 0,
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 σ(u)
and the quadratic nature of the Joule heating term forcing the Stefan
problem.
Finally, some numerical experiments are presented in two space dimensions.