Parametric finite element approximations of curvature driven interface evolutions

TL;DR

This paper presents efficient parametric finite element methods for simulating curvature-driven surface evolutions, ensuring stable computations with good mesh properties across various complex flows.

Contribution

It introduces novel computational techniques based on a weak formulation for mean curvature, improving mesh stability and applicability to multiple curvature-driven phenomena.

Findings

Methods demonstrate stability and mesh quality

Applicable to diverse curvature flows

Showcases effectiveness in biomembrane modeling

Abstract

Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very non-uniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.