A finite element error analysis for axisymmetric mean curvature flow

TL;DR

This paper provides an optimal error analysis for finite element approximations of axisymmetric mean curvature flow, confirming theoretical results through convergence experiments and numerical simulations on various surfaces.

Contribution

It introduces a rigorous finite element error analysis for axisymmetric mean curvature flow, including optimal bounds and numerical validation for genus-0 and genus-1 surfaces.

Findings

Optimal error bounds in $L^2$ and $H^1$ norms for genus-1 surfaces

Convergence experiments confirm theoretical error estimates

Numerical simulations demonstrate applicability to various surfaces including Angenent torus

Abstract

We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface, we derive optimal error bounds with respect to the $L^2$-- and $H^1$--norms for a fully discrete approximation. We perform convergence experiments to confirm the theoretical results, and also present numerical simulations for some genus-0 and genus-1 surfaces, including for the Angenent torus.