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arXiv · 2607.26290

Stabilization and optimal $L^2$ convergence of Dziuk's method with piecewise linear parametric finite elements for curve-shortening flow

Genming Bai
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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--N\"urnberg (BGN) type under the parabolic scaling $\tau\simeq h^2$---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 $L^2$ convergence.

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Genming Bai. 2026-07-28. Stabilization and optimal $L^2$ convergence of Dziuk's method with piecewise linear parametric finite elements for curve-shortening flow. https://arxiv.org/abs/2607.26290

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