Semi-discrete error estimates and implementation of a mixed method for the Stefan problem^∗

¹ Department of Mathematics, Tennessee Tech University, 1 William L Jones Dr, Cookeville, TN 38505, USA.
cbdavis@tntech.edu
² Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA.
walker@math.lsu.edu

Received: 29 July 2016
Revised: 19 January 2017
Accepted: 20 April 2017

Abstract

We analyze a dual formulation and finite element method for simulating the Stefan problem with surface tension (originally presented in [C.B. Davis and S.W. Walker, Int. Free Bound. 17 (2015) 427–464]). The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solid-liquid interface) as a constraint. The computational method uses a conforming mesh approach to accurately capture the jump conditions across the interface. Preliminary error estimates are derived, under reduced regularity assumptions, for the difference between the time semi-discrete solution and the fully discrete solution over one time step. Moreover, details of the implementation are discussed including mesh generation issues. Several simulations of interface growth (in two dimensions) are presented to illustrate the method.