A convergent evolving finite element algorithm for Willmore flow of closed surfaces

TL;DR

This paper introduces a new finite element algorithm for simulating Willmore flow of closed surfaces, providing a rigorous convergence proof and demonstrating optimal error bounds through numerical experiments.

Contribution

It presents a novel evolving surface finite element method for Willmore flow with a convergence proof and optimal error estimates, advancing numerical analysis of fourth-order surface evolution.

Findings

Proves convergence of the finite element scheme for Willmore flow.

Achieves optimal-order $H^1$-norm error bounds.

Numerical experiments confirm theoretical results.

Abstract

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix--vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.