A Stable Time Discretization of the Stefan Problem with Surface Tension

Abstract

We present a time discretization for the single phase Stefan problem with Gibbs--Thomson law. The method resembles an operator splitting scheme with an evolution step for the temperature distribution and a transport step for the dynamics of the free boundary. The evolution step involves only the solution of a linear equation that is posed on the old domain. We prove that the proposed scheme is stable in function spaces of high regularity. In the limit $\Delta t\to 0$ we find strong solutions of the continuous problem. This proves consistency of the scheme, and additionally it yields a new short-time existence result for the continuous problem.