Curve shortening flow coupled to lateral diffusion
Abstract
We present and analyze a semidiscrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of \cite{D99} for the curve shortening flow and \cite{DE07} for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in \cite{D99} an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.06173
 Bibcode:
 2015arXiv151006173P
 Keywords:

 Mathematics  Numerical Analysis;
 65M15;
 65M60;
 35K65
 EPrint:
 25 pages, 2 figures