Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary, to a reaction–diffusion equation on the curve

Abstract

We consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain Ω ⊂ R 2 , such that the curve meets the boundary 𝜕Ω orthogonally, and the forcing is a function of the solution of a reaction-diffusion equation that holds on the evolving curve. We prove optimal order H 1 error bounds for the resulting approximation and present numerical experiments.

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

“…For curve shortening flow coupled to a diffusion on a closed curve optimal-order finite element semi-discrete error estimates were shown in [47], while [8] have proved convergence of the corresponding backward Euler full discretisation. The case of open curves with a fix boundary was analysed in [49]. For forced-elastic flow of curves semi-discrete error estimates were proved in [48].…”