Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies

Abstract

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
for regularized crystalline surface energy densities.