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

by   Tim Binz, et al.

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.



There are no comments yet.


page 40

page 41

page 42


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

A proof of convergence is given for a novel evolving surface finite elem...

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

Optimal-order uniform-in-time H^1-norm error estimates are given for sem...

A convergent algorithm for mean curvature flow driven by diffusion on the surface

The evolution of a closed two-dimensional surface driven by both mean cu...

Error estimates for general non-linear Cahn-Hilliard equations on evolving surfaces

In this paper, we consider the Cahn-Hilliard equation on evolving surfac...

Qualitative and numerical aspects of a motion of a family of interacting curves in space

In this article we investigate a system of geometric evolution equations...

A filtered Boris algorithm for charged-particle dynamics in a strong magnetic field

A modification of the standard Boris algorithm, called filtered Boris al...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.