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° piecewise-linear in space Galerkin approximations. We find an L2 rate of convergence of order \^ in the e-regularization and an L rate of convergence of order (h2/e + Ai/ \6F) in the Galerkin estimates which leads to the natural choices £ ~ A4/3, A; ~ A4/3, and a resulting 0(/i2//3) L2 rate of convergence of the numerical scheme to the solution of the differential equation. An essentially 0(h) rate is demonstrated when e = 0 and A/ ~ h2 in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations. The latter result is consistent with computational experience.