A convergent finite element algorithm for mean curvature flow in arbitrary codimension

TL;DR

This paper presents a new finite element algorithm for simulating mean curvature flow of surfaces in any codimension, providing optimal error estimates and demonstrating convergence through numerical experiments.

Contribution

It introduces a convergent finite element method for mean curvature flow in arbitrary codimension with rigorous error analysis and practical numerical validation.

Findings

Optimal-order uniform-in-time $H^1$-norm error estimates

Convergence proven for polynomial degree ≥ 2 and BDF orders 2-5

Numerical experiments confirm theoretical results in codimension 2

Abstract

Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. 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. Numerical experiments in codimension 2 illustrate and complement our theoretical results.