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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2503.18505 |
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| _version_ | 1866913790489001984 |
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| author | Li, Meng Wang, Lining Wang, Yiming |
| author_facet | Li, Meng Wang, Lining Wang, Yiming |
| contents | Existing studies on the convergence of numerical methods for curvature flows primarily focus on first-order temporal schemes. In this paper, we establish a novel error analysis for parametric finite element approximations of genus-1 axisymmetric mean curvature flow, formulated using two classical second-order time-stepping methods: the Crank-Nicolson method and the BDF2 method. Our results establish optimal error bounds in both the L^2-norm and H^1-norm, along with a superconvergence result in the H^1-norm for each fully discrete approximation. Finally, we perform convergence experiments to validate the theoretical findings and present numerical simulations for various genus-1 surfaces. Through a series of comparative experiments, we also demonstrate that the methods proposed in this paper exhibit significant mesh advantages. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error analysis for temporal second-order finite element approximations of axisymmetric mean curvature flow of genus-1 surfaces Li, Meng Wang, Lining Wang, Yiming Numerical Analysis Existing studies on the convergence of numerical methods for curvature flows primarily focus on first-order temporal schemes. In this paper, we establish a novel error analysis for parametric finite element approximations of genus-1 axisymmetric mean curvature flow, formulated using two classical second-order time-stepping methods: the Crank-Nicolson method and the BDF2 method. Our results establish optimal error bounds in both the L^2-norm and H^1-norm, along with a superconvergence result in the H^1-norm for each fully discrete approximation. Finally, we perform convergence experiments to validate the theoretical findings and present numerical simulations for various genus-1 surfaces. Through a series of comparative experiments, we also demonstrate that the methods proposed in this paper exhibit significant mesh advantages. |
| title | Error analysis for temporal second-order finite element approximations of axisymmetric mean curvature flow of genus-1 surfaces |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.18505 |