A convergent finite element method with minimal deformation rate for mean curvature flow
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910023409467392 |
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| author | Huang, Tiantian Li, Buyang Tang, Rong |
| author_facet | Huang, Tiantian Li, Buyang Tang, Rong |
| contents | We propose and analyze a fully discrete parametric finite element method with minimal deformation rate (MDR) for simulating the mean curvature flow of general closed surfaces in three dimensions. The method is formulated from a coupled system that enforces the mean curvature flow law for the normal velocity while introducing an artificial tangential velocity that minimizes the deformation-rate energy, thereby preserving mesh quality without requiring remeshing or reparametrization. An $L^{2}$-projected averaged normal vector is used in the scheme to facilitate a rigorous convergence analysis. Within the projected--distance framework, we establish the first complete convergence proof for a parametric finite element method that incorporates the MDR tangential motion without relying on evolution equations for the mean curvature or the normal vector, achieving optimal-order error estimates for finite elements of degree $k \ge 3$. Numerical experiments corroborate the theoretical results and demonstrate that the proposed MDR method maintains mesh quality comparable to the Barrett--Garcke--Nürnberg method, for which convergence has not yet been established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14405 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A convergent finite element method with minimal deformation rate for mean curvature flow Huang, Tiantian Li, Buyang Tang, Rong Numerical Analysis 65M15, 65M60, 53E10, 35K65 We propose and analyze a fully discrete parametric finite element method with minimal deformation rate (MDR) for simulating the mean curvature flow of general closed surfaces in three dimensions. The method is formulated from a coupled system that enforces the mean curvature flow law for the normal velocity while introducing an artificial tangential velocity that minimizes the deformation-rate energy, thereby preserving mesh quality without requiring remeshing or reparametrization. An $L^{2}$-projected averaged normal vector is used in the scheme to facilitate a rigorous convergence analysis. Within the projected--distance framework, we establish the first complete convergence proof for a parametric finite element method that incorporates the MDR tangential motion without relying on evolution equations for the mean curvature or the normal vector, achieving optimal-order error estimates for finite elements of degree $k \ge 3$. Numerical experiments corroborate the theoretical results and demonstrate that the proposed MDR method maintains mesh quality comparable to the Barrett--Garcke--Nürnberg method, for which convergence has not yet been established. |
| title | A convergent finite element method with minimal deformation rate for mean curvature flow |
| topic | Numerical Analysis 65M15, 65M60, 53E10, 35K65 |
| url | https://arxiv.org/abs/2602.14405 |