"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

TL;DR

This paper introduces new diffuse approximation methods for the Willmore functional and flow, based on perimeter approximations, with theoretical convergence proofs and supporting numerical simulations.

Contribution

It presents novel diffuse approximations for the Willmore functional and flow, including Gamma-convergence analysis and asymptotic justification.

Findings

Gamma-convergence of the new approximation established

Convergence to Willmore flow justified via asymptotic expansion

Numerical simulations demonstrate effectiveness of the approximation

Abstract

We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [{\em Interfaces Free Bound. 14 (2012)}]. We identify the candidate for the $\Gamma$--convergence, prove the $\Gamma$--limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.