Tangential stability and fully discrete convergence of the classical BGN scheme for curve shortening flow
Abstract
We prove fully discrete convergence of the classical Barrett--Garcke--Nürnberg (BGN) scheme for curve-shortening flow of smooth embedded closed planar curves. The main obstruction is that the mass form controls only normal motion, whereas the tangential motion is determined implicitly by the curvature equation and governs the parametrization. The usual length-decay estimate therefore does not control perturbations of the full position update. We separate temporal and spatial errors through the time-semidiscrete BGN solution. A scalar normal resolvent and exact curvature and length identities yield uniform regularity and first-order time convergence. For the spatial analysis, an adapted normal--tangential norm gives a near-contractive estimate for the linearized update. Gauss--Lobatto cancellations and an exact covariance identity for the assembled nodal normals produce an H1 one-step defect of order hk+1, while an exact difference identity controls the nonlinear remainder. For every fixed k≥1, including the original piecewise linear method, we obtain the matched W1,∞ error bound C(τ+hk) on periodic quasi-uniform meshes with h≤ cτ2. All discrete steps are uniquely solvable, and the numerical curves remain regular and embedded. To the best of our knowledge, this is the first fully discrete convergence result for the classical BGN curve-shortening scheme without additional stabilization.
Related
- Harmonizable Multifractional Stable Motion: estimators for the Hurst function and the stability parameter from discrete-time observations, their rates of convergence and the associated central limit theorems
- Super-approximation and optimal L2 convergence of the stabilized BGN method with piecewise linear parametric finite elements for curve-shortening flow
- A New Approach to Strong Convergence II. The Classical Ensembles
- Uniform convergence of canonical discrete uniformization on equilateral approximations
- Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise
BibTeX
@misc{rao2026tangential,
title = {{Tangential stability and fully discrete convergence of the classical BGN scheme for curve shortening flow}},
author = {Qiqi Rao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21868},
eprint = {2609.21868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21868},
}
RIS
TY - UNPB
TI - Tangential stability and fully discrete convergence of the classical BGN scheme for curve shortening flow
AU - Qiqi Rao
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21868
UR - https://arxiv.org/abs/2609.21868
AB - We prove fully discrete convergence of the classical Barrett--Garcke--Nürnberg (BGN) scheme for curve-shortening flow of smooth embedded closed planar curves. The main obstruction is that the mass form controls only normal motion, whereas the tangential motion is determined implicitly by the curvature equation and governs the parametrization. The usual length-decay estimate therefore does not control perturbations of the full position update. We separate temporal and spatial errors through the time-semidiscrete BGN solution. A scalar normal resolvent and exact curvature and length identities yield uniform regularity and first-order time convergence. For the spatial analysis, an adapted normal--tangential norm gives a near-contractive estimate for the linearized update. Gauss--Lobatto cancellations and an exact covariance identity for the assembled nodal normals produce an H^1 one-step defect of order h^k+1, while an exact difference identity controls the nonlinear remainder. For every fixed k≥1, including the original piecewise linear method, we obtain the matched W^1,∞ error bound C(τ+h^k) on periodic quasi-uniform meshes with h≤ cτ^2. All discrete steps are uniquely solvable, and the numerical curves remain regular and embedded. To the best of our knowledge, this is the first fully discrete convergence result for the classical BGN curve-shortening scheme without additional stabilization.
ER -