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

TL;DR

This paper develops a finite element method for simulating elastic flow coupled with a diffusion process on a curve, providing theoretical error estimates and numerical validation in the one-dimensional graph setting.

Contribution

It introduces a novel finite element approach with operator splitting and penalty techniques for coupled elastic flow and diffusion on curves, with rigorous error analysis.

Findings

Numerical simulations confirm the theoretical convergence rates.

The method remains stable beyond the theoretically covered parameter regimes.

Error estimates are established for the coupled elastic flow and diffusion problem.

Abstract

A finite element approach to the elastic flow of a curve coupled with a diffusion equation on the curve is analysed. Considering the graph case, the problem is weakly formulated and approximated with continuous linear finite elements, which is enabled thanks to second-order operator splitting. The error analysis builds up on previous results for the elastic flow. To obtain an error estimate for the quantity on the curve a better control of the velocity is required. For this purpose, a penalty approach is employed and then combined with a generalised Gronwall lemma. Numerical simulations support the theoretical convergence results. Further numerical experiments indicate stability beyond the parameter regime with respect to the penalty term which is covered by the theory.