具有三重结点的多相穆林斯-塞克尔卡问题的结构保持有限元方法

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Tokuhiro Eto, Harald Garcke, Robert Nürnberg
{"title":"具有三重结点的多相穆林斯-塞克尔卡问题的结构保持有限元方法","authors":"Tokuhiro Eto, Harald Garcke, Robert Nürnberg","doi":"10.1007/s00211-024-01414-x","DOIUrl":null,"url":null,"abstract":"<p>We consider a sharp interface formulation for the multi-phase Mullins–Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins–Sekerka flow, demonstrate the capabilities of the introduced method.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions\",\"authors\":\"Tokuhiro Eto, Harald Garcke, Robert Nürnberg\",\"doi\":\"10.1007/s00211-024-01414-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider a sharp interface formulation for the multi-phase Mullins–Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins–Sekerka flow, demonstrate the capabilities of the introduced method.</p>\",\"PeriodicalId\":49733,\"journal\":{\"name\":\"Numerische Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerische Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00211-024-01414-x\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerische Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-024-01414-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

摘要

我们考虑了多相 Mullins-Sekerka 流动的尖锐界面公式。该流动的特点是曲线网络不断演化,使得曲线的总表面能降低,而封闭相的面积保持不变。利用变分公式,我们引入了一种完全离散的有限元方法。我们的离散化方法对运动界面进行了参数近似,而这种近似与用于体方程的离散化方法无关。可以证明该方案是无条件稳定的,并满足精确的体积守恒特性。此外,离散曲线上顶点的固有切向速度导致了顶点的渐近等分布,这意味着在实践中无需重网格化。几个数值示例,包括三相 Mullins-Sekerka 流的收敛实验,证明了所引入方法的能力。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions

A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions

We consider a sharp interface formulation for the multi-phase Mullins–Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins–Sekerka flow, demonstrate the capabilities of the introduced method.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
期刊介绍: Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers: 1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis) 2. Optimization and Control Theory 3. Mathematical Modeling 4. The mathematical aspects of Scientific Computing
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信