Second order in time finite element schemes for curve shortening flow and curve diffusion

TL;DR

This paper introduces and analyzes second order in time finite element schemes for curve shortening flow and curve diffusion, providing optimal error bounds and demonstrating second order convergence through numerical experiments.

Contribution

The paper develops new second order in time finite element methods for curve evolution problems, with proven optimal error bounds and improved mesh quality handling.

Findings

Optimal error bounds established for the schemes

Numerical experiments confirm second order convergence

Schemes effectively maintain mesh quality during evolution

Abstract

We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.