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：我们考虑了多相Mullins-Sekerka流动的尖锐界面形式。该流动的特征是曲线网络演化，使得曲线的总表面能减少，同时封闭相的面积得以保持。利用变分公式，我们引入了一种全离散有限元方法。我们的离散化方法对移动界面进行参数化逼近，该逼近独立于体相方程所使用的离散化。该方案可以证明是无条件稳定的，并且满足精确的体积守恒性质。此外，离散曲线上顶点的固有切向速度使得顶点渐近等距分布，这意味着在实践中无需重新网格化。包括三相Mullins-Sekerka流动收敛实验在内的多个数值算例，展示了所引入方法的能力。