Finite element error analysis for a system coupling surface evolution to diffusion on the surface

TL;DR

This paper develops and analyzes a finite element numerical scheme for simulating the coupled evolution of a surface and diffusion processes on it, providing optimal error bounds and practical validation.

Contribution

It introduces a finite element method for a surface evolution and diffusion system, proving optimal error estimates assuming smooth solutions.

Findings

Optimal error bounds in $L^$ and $L^2$ norms.

Numerical experiments confirm theoretical error estimates.

Application to simulate diffusion-induced grain boundary motion.

Abstract

We consider a numerical scheme for the approximation of a system that couples the evolution of a two--dimensional hypersurface to a reaction--diffusion equation on the surface. The surfaces are assumed to be graphs and evolve according to forced mean curvature flow. The method uses continuous, piecewise linear finite elements in space and a backward Euler scheme in time. Assuming the existence of a smooth solution we prove optimal error bounds both in $L^\infty(L^2)$ and in $L^2(H^1)$. We present several numerical experiments that confirm our theoretical findings and apply the method in order to simulate diffusion induced grain boundary motion.