Structure-preserving parametric finite element method for curve diffusion based on Lagrange multiplier approaches
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909294647050240 |
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| author | Garcke, Harald Jiang, Wei Su, Chunmei Zhang, Ganghui |
| author_facet | Garcke, Harald Jiang, Wei Su, Chunmei Zhang, Ganghui |
| contents | We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton's method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, as measured by the manifold distance, while simultaneously preserving the geometric structure of the curve diffusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13443 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure-preserving parametric finite element method for curve diffusion based on Lagrange multiplier approaches Garcke, Harald Jiang, Wei Su, Chunmei Zhang, Ganghui Numerical Analysis 65M60, 65M12, 35K55, 53C44 We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton's method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, as measured by the manifold distance, while simultaneously preserving the geometric structure of the curve diffusion. |
| title | Structure-preserving parametric finite element method for curve diffusion based on Lagrange multiplier approaches |
| topic | Numerical Analysis 65M60, 65M12, 35K55, 53C44 |
| url | https://arxiv.org/abs/2408.13443 |