A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

TL;DR

This paper introduces a convergent finite element algorithm for simulating mean curvature flow of closed surfaces, providing rigorous proof of convergence and optimal error bounds for the method.

Contribution

It develops a novel finite element scheme that discretizes geometric evolution equations and proves its convergence with optimal error estimates.

Findings

The method achieves optimal $H^1$-norm error bounds.

Convergence is proven for polynomial degree at least two and BDF orders two to five.

Numerical experiments confirm theoretical convergence rates.

Abstract

A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. 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.