Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918148483055616 |
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| author | Garcke, Harald Nürnberg, Robert Zhao, Quan |
| author_facet | Garcke, Harald Nürnberg, Robert Zhao, Quan |
| contents | 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 \cite{BGN07}. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves Garcke, Harald Nürnberg, Robert Zhao, Quan Numerical Analysis 65M60, 65M15, 65M12, 35R01 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 \cite{BGN07}. 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. |
| title | Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves |
| topic | Numerical Analysis 65M60, 65M15, 65M12, 35R01 |
| url | https://arxiv.org/abs/2503.23152 |