Curve shortening flow coupled to lateral diffusion

Abstract

We present and analyze a semi-discrete 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 Dziuk (SIAM J Numer Anal 36(6):1808---1830, 1999) for the curve shortening flow and Dziuk and Elliott (IMA J Numer Anal 27(2), 262---292, 2007) 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 Dziuk (SIAM J Numer Anal 36(6):1808---1830, 1999) 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.