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This paper considers and proposes some algorithms to compute the mean curvature flow under topological changes. Instead of solving the fully nonlinear partial differential equations based on the level set approach, we propose some minimization algorithms based on the phase field approach. It is well known that zero-level set of the Allen-Cahn equation approaches the mean curvature flow before the onset of the topological changes; however, there are few papers systematically studying the evolution of the mean curvature flow under the topological changes. There are three main contributions of this paper. First, in order to consider various random initial conditions, we design several benchmark problems with topological changes, and we find different patterns of the evolutions of the solutions can be obtained if the interaction length (width of the interface) is slightly changed, which is different from the problems without topological changes. Second, we propose an energy penalized minimization algorithm which works very well for these benchmark problems, and thus furthermore, for the problems with random initial conditions. Third, we propose a multilevel minimization algorithm. This algorithm is shown to be more tolerant of the unsatisfying initial guess when there are and there are no topological changes in the evolutions of the solutions.

Some algorithms for the mean curvature flow under topological changes