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Convergence of the original BGN method for mean curvature flow

  • Yifei Li
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Abstract

We prove that the original Barrett--Garcke--Nürnberg (BGN) method for mean curvature flow of curves converges in the manifold distance with order h2 for the time step τ=h2. The key new technique is backward error analysis, which stems from the numerical analysis of ordinary differential equations. By combining the expansions of the discrete velocity in space and time, we construct an approximation flow whose interpolation satisfies the BGN method with an improved defect, of order h4 with tangential part of order h6. This construction also identifies the tangential velocity of the BGN method and shows that τ=O(h2) is the correct relation between τ and h. With the improved defect, we then prove H1 superconvergence of order h4 of the numerical solution to the approximation flow using techniques of evolving surface finite element methods, which yields the convergence in the manifold distance. Numerical experiments confirm the predicted orders of defects.

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Cite

BibTeXRIS
BibTeX
@misc{li2026convergence,
  title = {{Convergence of the original BGN method for mean curvature flow}},
  author = {Yifei Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25402},
  eprint = {2609.25402},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25402},
}
RIS
TY  - UNPB
TI  - Convergence of the original BGN method for mean curvature flow
AU  - Yifei Li
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25402
UR  - https://arxiv.org/abs/2609.25402
AB  - We prove that the original Barrett--Garcke--Nürnberg (BGN) method for mean curvature flow of curves converges in the manifold distance with order h^2 for the time step τ=h^2. The key new technique is backward error analysis, which stems from the numerical analysis of ordinary differential equations. By combining the expansions of the discrete velocity in space and time, we construct an approximation flow whose interpolation satisfies the BGN method with an improved defect, of order h^4 with tangential part of order h^6. This construction also identifies the tangential velocity of the BGN method and shows that τ=O(h^2) is the correct relation between τ and h. With the improved defect, we then prove H^1 superconvergence of order h^4 of the numerical solution to the approximation flow using techniques of evolving surface finite element methods, which yields the convergence in the manifold distance. Numerical experiments confirm the predicted orders of defects.
ER  -