A structure preserving front tracking finite element method for the Mullins--Sekerka problem

TL;DR

This paper presents a new finite element method for the Mullins--Sekerka problem that preserves structure, guarantees stability and volume conservation, and is validated through numerical simulations demonstrating accuracy and practicality.

Contribution

It introduces a fully discrete, structure-preserving finite element scheme for the Mullins--Sekerka problem with proven stability and volume conservation.

Findings

Unconditionally stable numerical scheme

Exact volume conservation achieved

Effective for nearly crystalline surface energies

Abstract

We introduce and analyse a fully discrete approximation for a mathematical model for the solidification and liquidation of materials of negligible specific heat. The model is a two-sided Mullins--Sekerka problem. The discretization uses finite elements in space and an independent parameterization of the moving free boundary. We prove unconditional stability and exact volume conservation for the introduced scheme. Several numerical simulations, including for nearly crystalline surface energies, demonstrate the practicality and accuracy of the presented numerical method.