Space-time finite element methods for surface diffusion with applications to the theory of the stability of cylinders

A family of space-time finite element approximation schemes is presented for the nonlinear partial differential equations governing diffusion in the surface of a body of revolution. The schemes share with the partial differential equations properties of conservation of volume and decrease of area. Numerical experiments are described showing that the result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial Fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory. Examples are given of cases in which longitudinal perturbations with high wave-number spectra grow in amplitude, after an initial rapid decay followed by a long ''incubation period,'' and result in break-up of the body into a necklace of beads. The results of finite element calculations are compared with the predictions of a perturbation analysis.