A structure-preserving finite element method for the multi-phase Mullins-Sekerka problem with triple junctions

TL;DR

This paper introduces a novel finite element method for multi-phase Mullins-Sekerka flow that preserves structure, ensures stability, conserves volume, and maintains well-distributed vertices without remeshing.

Contribution

The paper presents a fully discrete, structure-preserving finite element scheme for multi-phase Mullins-Sekerka flow with triple junctions, featuring volume conservation and automatic vertex redistribution.

Findings

The method is unconditionally stable.

It conserves enclosed phase areas exactly.

Numerical examples demonstrate convergence and effectiveness.

Abstract

We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.