A Mesh-Free Solver for Multi-Phase Mullins-Sekerka Flow: Triple Junctions and Ninety-Degree Boundary Contact

Abstract

A bulk mesh-free solver for the multi-phase Mullins--Sekerka flow in $\mathbb R^2$ and in a half-plane bounded by a Neumann wall is developed. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. We use a charge simulation method, a variant of the method of fundamental solutions: each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh or singular integral is required. It treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary the no-flux condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving: between topological events, every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The junction connectivity is fixed, the only topological event being the disappearance of a region enclosed by a single closed curve and incident to no junction. The proposed scheme is validated against an exact three-concentric-circle solution in terms of convergence, cost, and conditioning.