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

TL;DR

The paper tackles the degenerate multi-phase Stefan problem with triple junctions by developing a parametric finite element method (PFEM) that tracks moving interfaces via unfitted, parametrically represented surfaces. It presents both a linear and a nonlinear structure-preserving PFEM, proves existence/uniqueness of the discrete solution and unconditional stability, and introduces a variant that conserves a discrete energy quantity. The approach leverages a gradient-flow perspective, leveraging the Dirichlet-to-Neumann operator to link interface velocity to curvature, and yields a coupled bulk–surface linear system that can be solved efficiently via Krylov methods or Schur-complement techniques. Extensive 2D and 3D numerical experiments demonstrate convergence against exact solutions, robustness to topology changes (including triple junctions), energy dissipation, and accurate reproduction of interface dynamics, establishing the method’s practicality for complex multi-phase surface evolution.

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.

Figures (15)

• Figure 1: A surface cluster in 2d made up of three open curves and two triple junctions.
• Figure 2: Example triangulation $\Gamma^m$ of a triple bubble with polygonal triple junction line $\mathcal{T}^m_1$, where here $Z_1 = 24$.
• Figure 3: Adaptive bulk mesh in two space dimensions.
• Figure 4: Two concentric circles.
• Figure 5: Comparison of the discrete (black, solid) and exact (red, dashed) radii for $(\beta_1,\beta_2,\beta_3) = (-1,0,1)$.

Theorems & Definitions (14)

• Proposition 2.1: Energy dissipation
• proof
• Proposition 2.2: Total energy content preservation
• proof
• Remark 2.1
• Theorem 4.1: Existence and uniqueness
• proof
• Theorem 4.2: Unconditional stability
• Theorem 4.2: Unconditional stability
• proof