Title: Space-time finite element methods for
surface diffusion with applications to
the theory of stability of cylinders
Authors: Bernard D. Coleman, Richard S. Falk
and Maher Moakher
Source: SIAM J. on Scientific Computing
Vol. 17 (1996), 1434-1448.
Status: Published
Abstract: 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.
Keywords: axially symmetric motion by Laplacian of mean
curvature, stability against surface diffusion
Subj. class.: 65M60, 73V05, 73T05