Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves

We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation originally proposed by Barrett, Garcke and Nürnberg (BGN) in (...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 105; no. 2; p. 45
Main Authors: Garcke, Harald, Nürnberg, Robert, Zhao, Quan
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2025
Springer Nature B.V
Subjects:
Algorithms
Apexes
Applied mathematics
Approximation
Computational Mathematics and Numerical Analysis
Curvature
Energy
Finite element method
Lagrange multiplier
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Finite element method
65M15
Planar curves
Willmore flow
Energy stability
Equidistribution
35R01
65M60
65M12
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation originally proposed by Barrett, Garcke and Nürnberg (BGN) in (Barrett et al. in J. Comput. Phys. 222:441–467, 2007). Under discretization in space with piecewise linear elements this leads to a stable continuous-in-time semidiscrete scheme, which retains the equidistribution property from the BGN methods. Furthermore, two fully discrete schemes can be shown to satisfy unconditional energy stability estimates. Numerical examples are presented to showcase the good properties of the introduced schemes, including an asymptotic equidistribution of vertices.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-03068-9