Curve shortening flow coupled to lateral diffusion
This work provides a theoretical foundation for numerical methods in coupled geometric and parabolic systems, which is incremental for researchers in computational geometry and PDEs.
The authors present and analyze a semi-discrete finite element scheme for curve shortening flow coupled with lateral diffusion, proving convergence with additional estimates for the time derivative of the length element error. Numerical simulations support the theoretical results.
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 \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.