Elastic flow interacting with a lateral diffusion process: The one-dimensional graph case
Provides a theoretical and numerical framework for a coupled geometric PDE problem, but is incremental as it extends existing elastic flow analysis.
The paper develops and analyzes a finite element method for elastic flow of a curve coupled with a lateral diffusion equation on the curve, proving error estimates and demonstrating convergence numerically.
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.