  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.
