A parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions

TL;DR

This paper introduces a parametric finite element method for modeling the complex energy-driven motion of multi-phase surfaces with triple junctions, ensuring stability, uniqueness, and structure preservation.

Contribution

It develops a novel unfitted finite element scheme for a degenerate multi-phase Stefan problem with proven stability and a structure-preserving variant.

Findings

The scheme guarantees existence and uniqueness of solutions.

Numerical results demonstrate the scheme's applicability.

A structure-preserving modification maintains key invariants.

Abstract

In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.