A convergent finite element algorithm for generalized mean curvature flows of closed surfaces

TL;DR

This paper introduces a finite element algorithm for generalized mean curvature flows of closed surfaces, providing error estimates and convergence proofs, and demonstrating its effectiveness through numerical experiments including non-convex surfaces.

Contribution

The paper develops a convergent finite element method for generalized mean curvature flows, including inverse and power mean curvature flows, with rigorous error analysis and stability proofs.

Findings

Optimal-order $H^1$-norm error bounds established

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

Numerical experiments confirm theoretical convergence and explore non-convex surfaces

Abstract

An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full discretisations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to non-linear second-order parabolic evolution equations for the normal velocity and normal vector. Convergence proofs are presented 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, normal velocity, and therefore for the mean curvature. The stability analysis is performed in the matrix-vector formulation, and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results, and also to report on monotone quantities, e.g.~Hawking mass for inverse mean curvature flow. Complemented by experiments for non-convex surfaces.