Error estimates for the multidimensional two-phase Stefan problem

In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and ${C^0}$ piecewise-linear in space Galerkin approximations. We find an ${L^2}$ rate of convergence of order $\sqrt \varepsilon$ in the $\varepsilon$-regularization and an ${L^2}$ rate of convergence of order $({h^2}/\varepsilon + \Delta t/\sqrt \varepsilon )$ in the Galerkin estimates which leads to the natural choices $\varepsilon \sim {h^{4/3}}$, $\Delta t \sim {h^{4/3}}$, and a resulting $O({h^{2/3}})\;{L^2}$ rate of convergence of the numerical scheme to the solution of the differential equation. An essentially $O(h)$ rate is demonstrated when $\varepsilon = 0$ and $\Delta t \sim {h^2}$ in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations. The latter result is consistent with computational experience.