Numerical Analysis for a System Coupling Curve Evolution to Reaction-Diffusion on the Curve
Provides rigorous numerical analysis for a coupled geometric PDE system, which is important for applications in materials science and biology, but the contribution is incremental as it extends existing techniques.
The paper develops and analyzes a finite element method for a system coupling curve evolution via forced curve shortening flow to a reaction-diffusion equation on the curve, proving optimal error bounds and demonstrating the benefit of tangential mesh motion.
We consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The scheme for the curve evolution is based on a parametric description allowing for tangential motion, whereas the discretisation for the PDE on the curve uses an idea from [6]. We prove optimal error bounds for the resulting fully discrete approximation and present numerical experiments. These confirm our estimates and also illustrate the advantage of the tangential motion of the mesh points in practice.