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Convergence of isoparametric BGN methods for mean curvature flow of closed surfaces with high-order finite elements

  • Yifei Li
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Abstract

We prove the convergence of the isoparametric Barrett--Garcke--Nürnberg (BGN) method for mean curvature flow of closed surfaces. For finite elements of degree k≥6 and time steps τ=O(h2), the numerical surfaces converge to the exact surfaces in the manifold distance with order h2. The proof relies on two ingredients. First, we construct approximation flows by an expansion in time, whose interpolations satisfy the BGN method with an improved defect. Second, we derive a refined estimate for the tangential error from an algebraic identity for tangential test functions.

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Cite

BibTeXRIS
BibTeX
@misc{li2026convergence,
  title = {{Convergence of isoparametric BGN methods for mean curvature flow of closed surfaces with high-order finite elements}},
  author = {Yifei Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08972},
  eprint = {2610.08972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08972},
}
RIS
TY  - UNPB
TI  - Convergence of isoparametric BGN methods for mean curvature flow of closed surfaces with high-order finite elements
AU  - Yifei Li
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08972
UR  - https://arxiv.org/abs/2610.08972
AB  - We prove the convergence of the isoparametric Barrett--Garcke--Nürnberg (BGN) method for mean curvature flow of closed surfaces. For finite elements of degree k≥6 and time steps τ=O(h^2), the numerical surfaces converge to the exact surfaces in the manifold distance with order h^2. The proof relies on two ingredients. First, we construct approximation flows by an expansion in time, whose interpolations satisfy the BGN method with an improved defect. Second, we derive a refined estimate for the tangential error from an algebraic identity for tangential test functions.
ER  -