On the stable discretization of strongly anisotropic phase field models with applications to crystal growth

TL;DR

This paper presents unconditionally stable finite element methods for anisotropic Allen--Cahn and Cahn--Hilliard equations, enabling reliable simulations in materials science with proven stability and demonstrated numerical effectiveness.

Contribution

It introduces the first fully practical, unconditionally stable finite element approximations for anisotropic phase field models, with rigorous stability proofs and numerical validation.

Findings

Proved unconditional stability of the proposed finite element schemes

Demonstrated effectiveness through numerical experiments

Applicable to phase field models in materials science

Abstract

We introduce unconditionally stable finite element approximations for anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations frequently feature in phase field models that appear in materials science. On introducing the novel fully practical finite element approximations we prove their stability and demonstrate their applicability with some numerical results. We dedicate this article to the memory of our colleague and friend Christof Eck (1968--2011) in recognition of his fundamental contributions to phase field models.