Stabilization and optimal L² convergence of Dziuk's method with piecewise linear parametric finite elements for curve-shortening flow

2026/07/28 by Genming Bai
Computer Science · Mathematics · #cs.NA #math.NA

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arxiv created 2026/07/28 · arxiv updated 2026/07/30

Abstract

We propose a stabilized version of the fully discrete Dziuk's method for the curve-shortening flow of a closed planar curve with piecewise linear parametric finite elements. With a carefully designed stabilization term, we are able to show a surprising discrete tangential stability of the Barrett--Garcke--Nürnberg (BGN) type under the parabolic scaling τ≃ h²---a feature hidden at the continuous level. Together with a new super-approximation result for the reversely averaged normal vector of linear elements, this Dziuk-type discrete tangential stability yields optimal L² convergence.

Citations

• A convergent finite element method for two-phase Stokes flow driven by surface tension
• Convergence analysis for the Barrett--Garcke--Nurnberg method of transport type for evolving curves
• A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow
• Convergence of Dziuk's semidiscrete finite element method for mean curvature flow of closed surfaces with high-order finite elements
• A convergent evolving finite element algorithm for mean curvature flow of closed surfaces
• Convergence of finite elements on an evolving surface driven by\n diffusion on the surface
• Numerical Analysis for a System Coupling Curve Evolution to Reaction-Diffusion on the Curve
• High-order evolving surface finite element method for parabolic problems on evolving surfaces
• On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick
• Curve shortening flow coupled to lateral diffusion
• Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow

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• Stabilization technique applied to curve shortening flow in ℝ³
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• Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves