  We propose and analyze volume-preserving parametric finite element methods
for surface diffusion, conserved mean curvature flow and an intermediate
evolution law in an axisymmetric setting. The weak formulations are presented
in terms of the generating curves of the axisymmetric surfaces. The proposed
numerical methods are based on piecewise linear parametric finite elements. The
constructed fully practical schemes satisfy the conservation of the enclosed
volume. In addition, we prove the unconditional stability and consider the
distribution of vertices for the discretized schemes. The introduced methods
are implicit and the resulting nonlinear systems of equations can be solved
very efficiently and accurately via the Newton's iterative method. Numerical
results are presented to show the accuracy and efficiency of the introduced
schemes for computing the considered axisymmetric geometric flows.
