A novel finite element approximation of anisotropic curve shortening flow

IF 1.2 4区 数学 Q1 MATHEMATICS
Klaus Deckelnick, Robert Nürnberg
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引用次数: 3

Abstract

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal $H^1$-error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.
各向异性曲线缩短流的一种新的有限元逼近
我们将DeTurck技巧从经典的各向同性曲线缩短流推广到各向异性环境。在这里,各向异性能量密度被允许依赖于空间,这允许在芬斯勒度量的背景下进行解释,例如,在黎曼流形中产生测地线曲率流。假设密度是严格凸光滑的,我们引入了一个新的各向异性曲线缩短流的弱公式。然后,我们导出了使用分段线性单元的连续半离散有限元近似的最优$H^1$-误差界。此外,我们考虑了一些完全实用的完全离散格式,并证明了它们的无条件稳定性。最后,我们给出了几个数值模拟,包括一些收敛实验,证实了推导的误差界限,以及在结晶曲率流和测地线曲率流中的应用。
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
17
审稿时长
>12 weeks
期刊介绍: Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems in all areas where such phenomena are pertinent. The journal aims to be a forum where mathematical analysis, partial differential equations, modelling, scientific computing and the various applications which involve mathematical modelling meet. Submissions should, ideally, emphasize the combination of theory and application.
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