A minimizing-movement framework for parametric finite element approximation of anisotropic curve shortening flows
Abstract
We propose a minimizing-movement framework for constructing parametric finite element methods (PFEMs) for anisotropic curve shortening flows. The cornerstone is an admissible modified energy: a smooth, convex, and iteration-dependent surrogate that majorizes the original anisotropic energy and agrees with it at the current iterate. For any fixed finite element degree n≥1 and any positive surface energy density γ∈ C2(𝕊1), the unconstrained one-step problem has a unique solution for any time-step size whenever the parametric derivative of the current discrete curve does not vanish on any mesh element. It also satisfies a one-step discrete energy dissipation inequality. It therefore preserves the variational structure of the underlying L2-gradient flow. The framework recovers a variational interpretation of Dziuk's isotropic scheme and applies directly to the weak anisotropies. For strong anisotropies, a convex splitting removes the weak-anisotropy restriction on the original density. A scalar constraint further extends the framework to area-preserving flows. Numerical experiments exhibit approximately the expected convergence for smooth isotropic and weakly anisotropic examples. They also confirm energy dissipation and area preservation. In the strongly anisotropic regime, the computations produce sharp facet-like profiles but reveal mesh-scale oscillations under the grid refinement.
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BibTeX
@misc{jiang2026minimizingmovement,
title = {{A minimizing-movement framework for parametric finite element approximation of anisotropic curve shortening flows}},
author = {Wei Jiang and Chunmei Su and Ganghui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36753},
eprint = {2610.36753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36753},
}
RIS
TY - UNPB
TI - A minimizing-movement framework for parametric finite element approximation of anisotropic curve shortening flows
AU - Wei Jiang
AU - Chunmei Su
AU - Ganghui Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36753
UR - https://arxiv.org/abs/2610.36753
AB - We propose a minimizing-movement framework for constructing parametric finite element methods (PFEMs) for anisotropic curve shortening flows. The cornerstone is an admissible modified energy: a smooth, convex, and iteration-dependent surrogate that majorizes the original anisotropic energy and agrees with it at the current iterate. For any fixed finite element degree n≥1 and any positive surface energy density γ∈ C^2(𝕊^1), the unconstrained one-step problem has a unique solution for any time-step size whenever the parametric derivative of the current discrete curve does not vanish on any mesh element. It also satisfies a one-step discrete energy dissipation inequality. It therefore preserves the variational structure of the underlying L^2-gradient flow. The framework recovers a variational interpretation of Dziuk's isotropic scheme and applies directly to the weak anisotropies. For strong anisotropies, a convex splitting removes the weak-anisotropy restriction on the original density. A scalar constraint further extends the framework to area-preserving flows. Numerical experiments exhibit approximately the expected convergence for smooth isotropic and weakly anisotropic examples. They also confirm energy dissipation and area preservation. In the strongly anisotropic regime, the computations produce sharp facet-like profiles but reveal mesh-scale oscillations under the grid refinement.
ER -