We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman et al. (Diffusion generated motion by mean curvature. Department of Mathematics, University of California, Los Angeles 1992). We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu et al. (Commun Pure Appl Math 68(5):808-864, 2015). This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movements scheme for an energy functional that -converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren et al. (SIAM J Control Optim 31(2):387-438, 1993) and Luckhaus and Sturzenhecker (Calculus Var Partial Differ Equ 3(2):253-271, 1995), which establish convergence of a more academic minimizing movements scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which however is not ensured by the compactness coming from the basic estimates.

We present a convergence result for solutions of the vector‐valued Allen‐Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multiphase mean~curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time‐integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and volume constraints.© 2018 Wiley Periodicals, Inc.

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In this paper we establish the convergence of three computational algorithms for interface motion in a multi-phase system, which incorporate bulk effects. The algorithms considered fall under the classification of thresholding schemes, in the spirit of the celebrated Merriman-BenceOsher algorithm for producing an interface moving by mean curvature. The schemes considered here all incorporate either a local force coming from an energy in the bulk, or a non-local force coming from a volume constraint. We first establish the convergence of a scheme proposed by Ruuth-Wetton for approximating volume-preserving mean-curvature flow. Next we study a scheme for the geometric flow generated by surface tension plus bulk energy. Here the limit is motion by mean curvature (MMC) plus forcing term. Last we consider a thresholding scheme for simulating grain growth in a polycrystal surrounded by air, which incorporates boundary effects on the solid-vapor interface. The limiting flow is MMC on the inner grain boundaries, and volume-preserving MMC on the solid-vapor interface.