Error estimates for the multidimensional two-phase Stefan problem

Abstract

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 {C^0} piecewise-linear in space Galerkin approximations. We find an L 2 {L^2} rate of convergence of order ε \sqrt \varepsilon in the ε \varepsilon -regularization and an L 2 {L^2} rate of convergence of order ( h 2 / ε + Δ t / ε ) ({h^2}/\varepsilon + \Delta t/\sqrt \varepsilon ) in the Galerkin estimates which leads to the natural choices ε ∼ h 4 / 3 \varepsilon \sim {h^{4/3}} , Δ t ∼ h 4 / 3 \Delta t \sim {h^{4/3}} , and a resulting O ( h 2 / 3 ) L 2 O({h^{2/3}})\;{L^2} rate of convergence of the numerical scheme to the solution of the differential equation. An essentially O ( h ) O(h) rate is demonstrated when ε = 0 \varepsilon = 0 and Δ t ∼ h 2 \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.