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.

翻译：我们考虑多相Mullins-Sekerka流的尖锐界面公式。该流动由曲线网络演化表征，其目标是减少曲线的总表面能，同时保持封闭相的面积守恒。基于变分公式，我们引入一种全离散有限元方法。我们的离散化方案对移动界面采用参数化逼近，该逼近独立于体相方程所用的离散化。该方案可证明是无条件稳定的，且满足精确的体积守恒性质。此外，离散曲线上顶点固有的切向速度可实现渐近等距分布，这意味着在实践中无需重新网格化。包括三相Mullins-Sekerka流的收敛性实验在内的多个数值算例，展示了所提方法的能力。