Parametric finite element approximations for anisotropic surface diffusion with axisymmetric geometry

We consider parametric finite element approximations for the anisotropic surface diffusion flow in an axisymmetric setting. Based on the anisotropy function, we introduce a symmetric positive definite matrix with a suitable stabilizing function. This then gives rise to two novel weak formulations for the axisymmetric flow in terms of the generating curve of the interface. By using piecewise linear elements in space and backward Euler in time, we discretize the weak formulations to obtain different approximating methods. These include a linear approximation which has good mesh properties and nonlinear approximations which can be shown rigorously to satisfy the volume preservation or the unconditional energy stability on the discrete level. Extensive numerical results are reported to demonstrate the accuracy and efficiency of the introduced methods for computing the axisymmetric flow.