Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) RungeKutta scheme are employed. Polynomial spaces of quadratic order are used. A numerical example with a change of topology is provided, and the order of convergence is studied on the Frank sphere example.