Phase change problems arise in anumber of important physical and industrial contexts such as process engineering and geophysics. These problems are challenging to solve numerically due to the presence of moving phase change interfaces. Adaptive methods that allow the mesh elements to move in time have been developed to efficiently solve these problems. The following example shows the method in action. Initially the domain is split into a solid region at the bottom and a liquid region at the top. A heat source causes melting to occur in the lower solid region forming a cicular phase front. This grows until it merges with the upper liquid region. The liquid region then continues to expand towards the bottom of the container eventually splitting the solid region in two. The mesh has no difficulty in adapting to the two phase change interfaces. Furthermore, the mesh movement algorithm effectively deals with the change in topology as the two interfaces merge and separate.

G. Beckett, J.A. Mackenzie and M.L. Robertson.

A Moving Mesh Finite Element Method for the Solution of Two-Dimensional Stefan Problems.
Journal of Computational Physics, vol. 168, No. 2, pages 500-518, 2001.

Moving mesh for stefan problem