What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under freeze-thaw cycles. Can anyone give a somewhat detailed explanation?



The classical two phase Stefan problems
there is a very large literature, you could start for example from a [textbook](https://books.google.nl/books/about/The_Stefan_Problem.html?id=l0-t-sjVCqYC&redir_esc=y); an overview of numerical methods is given [here](http://www.sciencedirect.com/science/article/pii/S0377042705003730):



> 
> In this paper, we present a critical comparison of the suitability of
>  several numerical methods, level set, moving grid and phase field
>  model, to address two well-known Stefan problems in phase
>  transformation studies: melting of a pure phase and diffusional
>  solid-state phase transformations in a binary system. Similarity
>  solutions are applied to verify the numerical results.
> 
> 
> 


