Convergence of a scheme for an elastic flow with tangential mesh movement

IF 0.6 4区数学 Q4 STATISTICS & PROBABILITY

Esaim-Probability and Statistics Pub Date : 2023-03-01 DOI:10.1051/m2an/2022091

Paola Pozzi, Bjoern Stinner

引用次数: 2

Abstract

Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of Deckelnick and Dziuk [ Math. Comp. 78 (2009) 645–671] we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical results that support the theory.

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网格切向运动弹性流格式的收敛性

闭合曲线的弹性流动可能包含显著的变形。基于网格的近似方案需要切向重新分配顶点以进行长时间的计算。我们提出并分析了一种利用狄利克雷能量来达到这一目的的方法。该方法还对曲线的长度进行了有效的惩罚，并且平衡形状等效于弹性能量随长度泛函的增宽的驻点。我们的数值方法是基于线性参数有限元。按照Deckelnick和Dziuk[数学]的思路。Comp. 78(2009) 645-671]我们证明了收敛性并建立了误差估计，注意到与长度泛函相比，Dirichlet能量的加入简化了分析。我们还给出了一个简单的半隐式时间离散，并讨论了一些支持该理论的数值结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Probability and Statistics STATISTICS & PROBABILITY-

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original research and survey papers in the area of Probability and Statistics. It covers theoretical and practical aspects, in any field of these domains. Of particular interest are methodological developments with application in other scientific areas, for example Biology and Genetics, Information Theory, Finance, Bioinformatics, Random structures and Random graphs, Econometrics, Physics. Long papers are very welcome. Indeed, we intend to develop the journal in the direction of applications and to open it to various fields where random mathematical modelling is important. In particular we will call (survey) papers in these areas, in order to make the random community aware of important problems of both theoretical and practical interest. We all know that many recent fascinating developments in Probability and Statistics are coming from "the outside" and we think that ESAIM: P&S should be a good entry point for such exchanges. Of course this does not mean that the journal will be only devoted to practical aspects.