  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.
