Topological changes in surface evolution using Lagrangian approach

In this work, we discuss several methods that can solve topological changes during the Lagrangian surface evolution. The Lagrangian approach, contrarily to level-set methods, solves the surface evolution problem in a two-dimensional setting, but the processes of cutting and gluing the evolving surfaces are non-trivial. We present examples of mean curvature flow and an evolution by a constant normal speed. We made approximation of the mean curvature flow by a finite volume method -- the so called cotangent scheme. Then we suggest two methods that solve the most common topological changes."