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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月26日

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  • Harald Garcke,
  • Robert Nürnberg,
  • Quan Zhao

摘要

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.