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

Tokuhiro Eto, Harald Garcke, Robert Nürnberg

TL;DR

The paper addresses numerical approximation of sharp-interface multi-phase Mullins-Sekerka flow with triple junctions by introducing a variational, fully discrete parametric finite element method (BGN-type). The scheme is unconditionally stable and preserves two structural properties critical to the model: the weighted interfacial length dissipates as energy decreases, and the enclosed phase areas are exactly conserved, with a nonlinear variant achieving exact area preservation at the fully discrete level. A key feature is the tangential motion of vertices, which yields asymptotically equidistributed meshes and eliminates the need for remeshing. The authors generalize the approach to arbitrary multi-phase systems, provide convergence tests, and showcase extensive 3- and 4-phase evolutions under equal and unequal surface energies, including topology-changing scenarios, demonstrating robustness and practical applicability for complex curve networks.

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.

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

TL;DR

The paper addresses numerical approximation of sharp-interface multi-phase Mullins-Sekerka flow with triple junctions by introducing a variational, fully discrete parametric finite element method (BGN-type). The scheme is unconditionally stable and preserves two structural properties critical to the model: the weighted interfacial length dissipates as energy decreases, and the enclosed phase areas are exactly conserved, with a nonlinear variant achieving exact area preservation at the fully discrete level. A key feature is the tangential motion of vertices, which yields asymptotically equidistributed meshes and eliminates the need for remeshing. The authors generalize the approach to arbitrary multi-phase systems, provide convergence tests, and showcase extensive 3- and 4-phase evolutions under equal and unequal surface energies, including topology-changing scenarios, demonstrating robustness and practical applicability for complex curve networks.

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.
Paper Structure (20 sections, 9 theorems, 90 equations, 11 figures, 2 tables)

This paper contains 20 sections, 9 theorems, 90 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical properties
  3. Weak formulation
  4. Finite element approximation
  5. Solution of the linear system
  6. Obtaining a fully discrete area conservation property
  7. Generalization to multi-component systems
  8. Problem setting
  9. Weak formulation
  10. Finite element approximations
  11. Numerical results
  12. Convergence experiment
  13. Evolutions with equal surface energies
  14. 3 phases
  15. 4 phases
  16. ...and 5 more sections

Key Result

Proposition 2.1

Assume that $(\hbox{\boldmath{$w$}},\{\Gamma^{}_{}(t)$$\}_{0\leq t\leq T})$ is a solution to eq:StrongForm. Then it holds that where we have defined the weighted length $|\Gamma(t)|_\sigma = \sum_{i=1}^3 \sigma_i|\Gamma^{}_{i}(t)| = \sum_{i=1}^3 \sigma_i \int_{\Gamma^{}_{i}(t)} 1 \,\mathrm{d}\mathcal{H}^{1}$. Here $\,\mathrm{d}\mathcal{L}^{2}$ and $\,\mathrm{d}\mathcal{H}^{1}$ refer to integratio

Figures (11)

  • Figure 1: Three open curves with two triple junctions.
  • Figure 2: A network of curves which consists of three concentric circles.
  • Figure 3: The solution at times $t=0, 0.4, 0.8, 1$, and a plot of the discrete energy over time. Below we show the adaptive bulk mesh at times $t=0$ and $t=1$.
  • Figure 4: The solution at times $t=0, 4, 4.5, 6$, and a plot of the discrete energy over time.
  • Figure 5: The solution at times $t=0, 4, 5, 7$, and a plot of the discrete energy over time.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Proposition 2.1: Curve shortening property of strong solutions
  • proof
  • Proposition 2.2: Area preserving property of strong solutions
  • proof
  • Theorem 4.1: Existence and uniqueness
  • proof
  • Lemma 4.2
  • Theorem 4.3: Unconditional stability
  • proof
  • Remark 4.4
  • ...and 11 more