A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling

TL;DR

This paper introduces a stable, fully discrete finite element method for simulating the evolution of anisotropic multi-phase interfaces with kinetic undercooling, applicable to complex phenomena like ice crystal growth.

Contribution

It develops a novel unfitted finite element approach for anisotropic multi-phase Mullins-Sekerka problems, allowing independent interface approximation and demonstrating stability and versatility.

Findings

Method is unconditionally stable.

Capable of modeling multiple ice crystal junctions.

Numerical examples validate the approach.

Abstract

We consider a sharp interface formulation for an anisotropic multi-phase Mullins-Sekerka problem with kinetic undercooling. The flow is characterized by a cluster of surfaces evolving such that the total surface energy plus a weighted sum of the volumes of the enclosed phases decreases in time. Upon deriving a suitable variational formulation, we introduce a fully discrete unfitted finite element method. In this approach, the approximations of the moving interfaces are independent of the triangulations used for the equations in the bulk. Our method can be shown to be unconditionally stable. Several numerical examples demonstrate the capabilities of the introduced method. In particular, it is demonstrated that the evolution of multiple ice crystals with junctions can be modeled using the proposed approach.