[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83775-en":3,"doc-seo-83775-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83775,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","A Revisit to DeTurck Method on Curve Shortening Flow with Optimal Error Analysis","Curve shortening–DeTurck flow rewrites curve shortening flow through the DeTurck reparameterization, enabling improved numerical behavior for evolving planar curves. For discrete approximations, while H1 error bounds are known, sharp L2 optimality has remained unresolved. This paper establishes optimal L2 error estimates for linearized Euler and Crank–Nicolson time discretizations combined with finite elements of degree k≥1. An extrinsic construction of the curve shortening–DeTurck flow is provided, and numerical experiments confirm spatial and temporal convergence rates and mesh distribution properties.","arXiv :2607 .04105v1 [math .NA] 5 Jul 2026  \nA REVISIT TO DETURCK METHOD ON CURVE SHORTENING FLOW WITH  \nOPTIMAL ERROR ANALYSIS  \nBEIPING DUAN  \nAbstract. The curve shortening–DeTurck flow introduced by Elliott and Fritz (IMA J. Numer.  \nAnal. 37(2): 543–603, 2017) is a reparameterization of the curve shortening flow via the DeTurck trick. For corresponding discrete schemes, although the H 1-error estimate has been established, the optimal L2-error estimate remains open due to technical difficulties. In this paper, we prove optimal L2-error estimates for linearized Euler and Crank–Nicolson discretizations combined with finite elements of degree k ≥ 1 in space. Moreover, we provide an extrinsic approach to derive the curve shortening–DeTurck flow. Numerical experiments are presented to illustrate the mesh distribution properties and to verify the convergence rates in both space and time.  \n1. Introduction  \nSuppose Γ(t) is a closed regular planar curve which evolves with time and is parameterized by the diffeomorphism ⃗X(t, s) : I → Γ(t) with I a periodic interval. The evolving curve Γ(t) then can be described by virtue of the flow map ⃗X(t, s):  \n(1.1) ~~ ~~∂⃗X∂(tt,~~ ~~s) = ⃗v ◦ ⃗X(t, s),  \nwhere in the case of curve shortening flow (CSF) ⃗v = ∂2γId| Γ(t) with Id the identity map from Γ(t) onto itself and γ its arc length parameter. If we introduce the perimeter L = RΓ(t) 1dγ as the functional energy, then the CSF can be considered as the L2 gradient flow.  \nUtilizing the chain rule one can reform the CSF in the parameter space I base on which the weak form reads: Find ⃗X(t, s) ∈ H1 (I)2 with ⃗X(0, s) = ⃗X0 (s) such that  \n(1.2) ZI ∂∂⃗Xt · ⃗ξ 􀀌 ∂s⃗X 􀀌 ds = ZI ∂s􀀌sX·~~ ~~⃗∂􀀌s~~ ~~⃗ξds = 0 ∀⃗ξ ∈ H1 (I)2 .  \nApplying linearized Euler scheme in time (dealing with |∂s⃗X| explicitly) and piecewise linear elements in space one can get Dziuk’s original method [16] .  \nThe first attempt at an error analysis of Dziuk’s scheme was carried out in [17], where the H 1 error of a semidiscrete formulation for the CSF was studied. The H 1 error for the fully discrete scheme was established by [25] . We also note that prior to their work, [21] derived the L2-error estimate for the fully discrete scheme using finite elements of degree k ≥ 3. Subsequently, [22, 1] proved the convergence of the semidiscrete scheme for closed surfaces using finite elements of degreek ≥ 6, where the matrix-vector formulation developed in [20, 19] was employed.  \nDziuk’s method is simple to implement and leads to a decoupled system in the x-and y-directions (and z-direction if applicable) . However, the drawback is that it includes only normal velocities, which may cause vertices on the numerical curve to collapse into each other, so that 􀀌∂s⃗Xnh􀀌 —the discrete analogue of 􀀌 ∂s⃗X(t)􀀌 at t = tn—approaches 0 . For the CSF, an early method incorporating  \nKey words and phrases. curve shortening flow, DeTurck method, harmonic map heat flow, error analysis.  \n2 BEIPING DUAN  \ntangential motion was presented by [7], where the weak form reads: Find ⃗X(t, s) ∈ H 1 (I)2 with ⃗X(0, s) = ⃗X0 (s) such that  \n(1.3) ZI ∂∂⃗Xt · ⃗ξ 􀀌 ∂s⃗X 􀀌 2 ds + ZI ∂s⃗X · ∂s⃗ξds = 0 ∀⃗ξ ∈ H1 (I)2 .  \nThe spatial H 1 error of the semidiscrete scheme for (1.3) was also given in their paper. To the best  \nof our knowledge, the optimal L2 error of a fully discrete scheme based on (1.3) was first proved by  \n[11], published 30 years after [7] . Some of the tricks they used in their proof provided the original inspiration for the error estimates in this paper.  \n[4] presented a framework (referred as the Barrett-Garcke-N¨urnberg, BGN scheme) for inducing tangential motions for (1.1) with ⃗v · ⃗ν = f (H), where H and ⃗ν are the curvature and the unit normal, respectively. When applied to the CSF, this method is linear and preserves the energydecaying property. Moreover, it also enjoys good mesh distribution and has been widely used since then. Later, the idea was extended to anisotropic case in [5] . Ho","cbCaiaEOW6MW6vxr","https://ap.wps.com/l/cbCaiaEOW6MW6vxr","pdf",769905,5,1,22,"English","en",105,"# Introduction\n## Historical background and related error analyses\n## Tangential motion methods and BGN framework\n## DeTurck reparameterization and harmonic map ideas","[{\"question\":\"What is the curve shortening–DeTurck flow and why is it used?\",\"answer\":\"It is obtained by reparameterizing the curve shortening flow using the DeTurck trick, introducing a suitable tangential velocity. This helps maintain numerical mesh quality while the curve evolves.\"},{\"question\":\"Which discretization methods are analyzed in the paper for optimal L2 error estimates?\",\"answer\":\"The paper proves optimal L2 error estimates for linearized Euler and Crank–Nicolson discretizations in time, coupled with finite elements of degree k≥1 in space.\"},{\"question\":\"How do the numerical experiments support the theoretical results?\",\"answer\":\"They verify convergence rates in both space and time and illustrate mesh distribution properties, demonstrating the effectiveness of the curve shortening–DeTurck framework.\"}]",1784190333,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-revisit-to-deturck-method-on-curve-shortening-flow-with-optimal-error-analysis","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-revisit-to-deturck-method-on-curve-shortening-flow-with-optimal-error-analysis/83775/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the curve shortening–DeTurck flow and why is it used?","Question",{"text":76,"@type":77},"It is obtained by reparameterizing the curve shortening flow using the DeTurck trick, introducing a suitable tangential velocity. This helps maintain numerical mesh quality while the curve evolves.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Which discretization methods are analyzed in the paper for optimal L2 error estimates?",{"text":81,"@type":77},"The paper proves optimal L2 error estimates for linearized Euler and Crank–Nicolson discretizations in time, coupled with finite elements of degree k≥1 in space.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the numerical experiments support the theoretical results?",{"text":85,"@type":77},"They verify convergence rates in both space and time and illustrate mesh distribution properties, demonstrating the effectiveness of the curve shortening–DeTurck framework.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]