{"title":"An Energy-Stable Parametric Finite Element Method for the Planar Willmore Flow","authors":"Weizhu Bao, Yifei Li","doi":"10.1137/24m1633893","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 103-121, February 2025. <br/> Abstract. We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first involves the coupling of the outward unit normal vector [math] and the normal velocity [math], and the second concerns the time derivative of the mean curvature [math]. Based on these, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate our PFEM’s efficiency and validate its unconditional energy stability.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/24m1633893","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
Abstract
SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 103-121, February 2025. Abstract. We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first involves the coupling of the outward unit normal vector [math] and the normal velocity [math], and the second concerns the time derivative of the mean curvature [math]. Based on these, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate our PFEM’s efficiency and validate its unconditional energy stability.
期刊介绍：
SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.
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