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 $H^1$ one-step defect of order $h^{k+1}$, while an exact difference identity controls the nonlinear remainder. For every fixed $k\ge1$, including the original piecewise linear method, we obtain the matched $W^{1,\infty}$ error bound $C(τ+h^k)$ on periodic quasi-uniform meshes with $h\le cτ^2$. 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.