[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83540-en":3,"doc-seo-83540-105":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},83540,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Overlapping Domain Decomposition for Meshless Finite Difference Methods","Schwarz type domain decomposition methods require a partition of unity to merge subdomain solutions, yet mesh-based discretizations often use minimal or no overlap. This study examines, from a meshless perspective, how partition-of-unity continuity influences the algebraic Schwarz method when Poisson and Stokes problems are discretized via radial basis function finite differences (RBF-FD). Numerical experiments show that small overlaps reduce iteration counts and improve performance, eliminating the need for disjoint partitioning strategies.","arXiv :2607 .00842v1 [math .NA] 1 Jul 2026  \nOverlapping Domain Decomposition for Meshless Finite Difference Methods  \nAlexander Westermann [0009−0000−8452−9400], Oleg Davydov [0000−0001−8813−9485], and Stefan Turek [0000−0002−9740−6087]  \nAbstract Schwarz type domain decomposition methods generally require a partition of unity to combine solutions on subdomains. However, in mesh-based methods it is common to organize subdomains with minimal overlap, if any, which is facilitated by the availability of a mesh. This study analyzes how the continuity of the partition of unity affects the algebraic Schwarz method for Poisson and Stokes equations from ameshless point of view, whereby the underlying differential operators are discretized using the radial basis function finite difference (RBF-FD) method. We demonstrate numerically that, in this setting, small overlaps improve the performance of the domain decomposition, leading to smaller iteration counts, and therefore no disjoint partitioning technique is required.  \nKeywords Schwarz method, partition of unity, meshless methods, RBF-FD, Poisson equation, Stokes equations  \n1 Introduction  \nSchwarz’s domain decomposition going back to [8] is used in modern mesh-based discretization methods either as a stand-alone iterative method for solving linear systems, or as a preconditioner. In particular, it is applied to saddle point problems, such as the Stokes or Navier-Stokes equations, for which no classical iterative methods can be used due to the matrix structure [6, 10] .  \nAlexander Westermann  \nDepartment of Mathematics, JLU Giessen, e-mail: alexander.westermann@math.uni-giessen.de  \nOleg Davydov  \nDepartment of Mathematics, JLU Giessen, e-mail: [oleg.davydov@math.uni-giessen.de](oleg.davydov@math.uni-giessen.de)[ ](oleg.davydov@math.uni-giessen.de)Stefan Turek  \nInstitute for Applied Mathematics, TU Dortmund University, e-mail: Stefan.Turek@math.tu[dortmund.de](dortmund.de)  \n2 Alexander Westermann, Oleg Davydov, and Stefan Turek  \nFurthermore, numerous papers on finite elements, for example [6, 7, 13], have discussed the application of the additive Schwarz method as a smoother in multigrid methods, with a focus on decomposing into very small subdomains (batch- or cellbased) . These methods originate from the work of Vanka [11], which employed asymmetric coupled Gauss-Seidel smoother that can be identified as an algebraic Schwarz method.  \nIn this paper we explore the application of the additive Schwarz method to linear systems arising from a meshless finite difference method on irregular nodes. The main question is how to design subdomains without the help of a mesh, and we demonstrate that a crucial parameter is the amount of the overlap between different subdomains. In mesh-based methods it is common [5] to use subdomains that build a disjoint partition of the full computation domain. However, our experiments show that this is undesirable for the meshless methods.  \nThe second question we address is the influence of the choice of the partition of unity (PoU) used to combine subsolutions when subdomains overlap. It turns out that both versions of the algebraic additive Schwarz method introduced in [1], the restricted additive Schwarz method (RAS) and the additive Schwarz method with harmonic extension (ASH) perform well in the meshless setting if PoU is continuous, but ASH typically outperforms RAS with the simpler discontinuous PoU.  \nIn our numerical experiments we discretize a Poisson problem in 1D and 2D anda Stokes problem in 2D by using the radial basis function finite difference (RBF-FD) method with polyharmonic RBF and a polynomial extension, see for example [4] .  \nThe paper is organized as follows. After a brief Section 2 that introduces the two versions of the iterative Schwarz methods RAS and ASH studied in this paper in the meshless context, we present in Section 3 our numerical investigation of their performance in conjunction with two types of PoU for three example","cbCaid5VY3JGi2Jt","https://ap.wps.com/l/cbCaid5VY3JGi2Jt","pdf",1086331,5,1,12,"English","en",105,"# Introduction\n# Schwarz methods","[{\"question\":\"What problem does the paper address in overlapping domain decomposition for meshless methods?\",\"answer\":\"It studies how to design subdomains and overlap without relying on a mesh when discretizing Poisson and Stokes equations using the RBF-FD meshless method.\"},{\"question\":\"How does partition of unity continuity affect the algebraic Schwarz method?\",\"answer\":\"The paper compares versions of algebraic additive Schwarz methods (RAS and ASH) and shows they perform well in the meshless setting when the partition of unity is continuous.\"},{\"question\":\"Why do small overlaps improve performance, and is a disjoint partition required?\",\"answer\":\"Numerical results indicate that small overlaps lead to smaller iteration counts, so disjoint partitioning techniques are unnecessary for the considered meshless setting.\"}]",1784188701,30,{"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},"overlapping-domain-decomposition-for-meshless-finite-difference-methods","",{"@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/overlapping-domain-decomposition-for-meshless-finite-difference-methods/83540/",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 problem does the paper address in overlapping domain decomposition for meshless methods?","Question",{"text":76,"@type":77},"It studies how to design subdomains and overlap without relying on a mesh when discretizing Poisson and Stokes equations using the RBF-FD meshless method.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does partition of unity continuity affect the algebraic Schwarz method?",{"text":81,"@type":77},"The paper compares versions of algebraic additive Schwarz methods (RAS and ASH) and shows they perform well in the meshless setting when the partition of unity is continuous.",{"name":83,"@type":74,"acceptedAnswer":84},"Why do small overlaps improve performance, and is a disjoint partition required?",{"text":85,"@type":77},"Numerical results indicate that small overlaps lead to smaller iteration counts, so disjoint partitioning techniques are unnecessary for the considered meshless 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