We present a variational formulation for the evolution of surface clusters in ℝ3 by mean curvature ﬂow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient ﬂows, and present weak formulations of these ﬂows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric ﬁnite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature ﬂow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.
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Harald Garcke, Robert Nürnberg, John W. Barrett, Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies. Interfaces Free Bound. 12 (2010), no. 2, pp. 187–234DOI 10.4171/IFB/232