Error estimates for a surface finite element method for anisotropic mean curvature flow

TL;DR

This paper develops and proves error estimates for a finite element method simulating anisotropic mean curvature flow on surfaces, demonstrating convergence and providing numerical validation.

Contribution

It introduces a new finite element approach for anisotropic mean curvature flow and establishes its optimal convergence in the $H^1$-norm.

Findings

Proved convergence of the method in the $H^1$-norm

Achieved optimal-order error estimates for degree ≥ 2 finite elements

Numerical experiments confirm theoretical results

Abstract

Error estimates are proved for an evolving surface finite element semi-discretization for anisotropic mean curvature flow of closed surfaces. For the geometric surface flow, a system coupling the anisotropic evolution law to parabolic evolution equations for the surface normal and normal velocity is derived, which then serve as the basis for the proposed numerical method. The algorithm for anisotropic mean curvature flow is proved to be convergent in the $H^1$-norm with optimal-order for finite elements of degree at least two. Numerical experiments are presented to illustrate and complement our theoretical results.