Abstract
We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke’s notion of mean curvature flow. Our proof is based on a localized version of the minimizing movements interpretation of Esedoğlu and the second author. We apply De Giorgi’s variational interpolation to the thresholding scheme and pass to the limit in the resulting energy-dissipation inequality. The result is conditional in the sense that we assume the time-integrated energies of the approximations to converge to those of the limit.
Highlights
The thresholding scheme is a time discretization for mean curvature flow
It was realized that thresholding preserves the gradient-flow structure of mean-curvature flow in the sense that it can be viewed as a minimizing movements scheme for an energy that -converges to the total interfacial area
Selim Esedoglu and the second author [15] showed that thresholding preserves the gradient-flow structure of mean curvature flow in the sense that it can be viewed as a minimizing movements scheme χ n = arg min u
Summary
The thresholding scheme is a time discretization for mean curvature flow. Its structural simplicity is intriguing to both applied and theoretical scientists. Merriman et al [27] introduced the algorithm in 1992 to overcome the numerical difficulty of multiple scales in phase-field models. Their idea is based on an operator splitting for the Allen–Cahn equation, alternating between linear diffusion and thresholding. The latter replaces the fast reaction coming from the nonlinearity, i.e., the reaction-term, in the Allen–Cahn equation. Large-scale simulations [11,12,13] demonstrate the efficiency of a slight modification of the scheme.
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