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We present a variational formulation for the evolution of surface clusters in R3by mean curvature flow, surface diffusion and their anisotropic variants. We introducethe triple junction line conditions that are induced by the considered gradientflows, and present weak formulations of these flows. In addition, we consider thecase where a subset of the boundaries of these clusters are constrained to lie onan external boundary. These formulations lead to unconditionally stable, fully discrete,parametric finite element approximations. The resulting schemes have verygood properties with respect to the distribution of mesh points and, if applicable,volume conservation. This is demonstrated by several numerical experiments, includingisotropic double, triple and quadruple bubbles, as well as clusters evolvingunder anisotropic mean curvature flow and anisotropic surface diffusion, includingfor regularized crystalline surface energy densities.

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies