Variational discretization of axisymmetric curvature flows

We present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail. Although numerical methods for geometric flows have been used frequently in axisymmetric settings, numerical analysis results so far are rare. In this paper, we present stability, equidistribution, existence and uniqueness results for the introduced approximations. Numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one-dimensional problem has to be solved. The good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries.