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 [12] for the curve shortening flow and [13] 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 [12] 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.

Numerical Analysis for a System Coupling Curve Evolution to Reaction Diffusion on the Curve

“…Clearly, (1.1a) holds with the additional property that the velocity vector x t points in the normal direction. Coupling (1.5) to the PDE satisfied byw, Pozzi and Stinner have derived and analysed in [13] a finite element scheme for (1.1a,b) (with g ≡ 0) based on continuous piecewise linears.…”

A simple scheme for the approximation of self-avoiding inextensible curves

“…In recent time, gradient flows have received much attention, with respect to rigorous analysis, see Dziuk et al [12] as well as regarding discretization aspects, see Deckelnick and Dziuk [11], Barrett et al [2], Bartels [3], Dall'Acqua et al [10], Pozzi and Stinner [27].…”

Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

“…For coupled problems such as (1.1)-(1.3) we are not aware of any convergence results. Schemes for curve shortening flow instead of the above elastic flow have been analysed in [39] (semi-discrete) and [2] (fully discrete). The related work of [30] covers the case of a (weighted) H 1 flow instead an L 2 flow of the surface energy.…”

“…In order to proceed we need to analyse the error between c and c h . We here basically follow the lines of [39], Lemma 4.2. But we need to provide all details as the treatment of the terms with the time derivative of the length element is different here.…”