[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84598-en":3,"doc-seo-84598-105":30,"detail-sidebar-cat-0-en-105":96},{"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},84598,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Hybridizable Staggered Discontinuous Galerkin Methods for Polyharmonic Equations on Polytopes","Hybridizable staggered discontinuous Galerkin methods are constructed for arbitrary-order polyharmonic equations (−∆)^m u = f on shape-regular polytopal meshes in Rd, for any m ≥ 1, d ≥ 2, and polynomial degree k ≥ 0. A mixed variable σ = ∇^m u and a staggered primal–dual mesh enforce complementary continuity between scalar and tensor unknowns. Local trace and bubble enrichments stabilize low-order tensor spaces, yielding an equivalent stabilization-free weak Galerkin formulation via hybridization. Well-posedness and optimal energy error estimates are proved, and numerical experiments on polygonal and tetrahedral meshes confirm the predicted convergence rates.","arXiv :2607 .00831v1 [math .NA] 1 Jul 2026  \nHYBRIDIZABLE STAGGERED DISCONTINUOUS GALERKIN METHODS FOR POLYHARMONIC EQUATIONS ON POLYTOPES  \nLONG CHEN, XUEHAI HUANG, YULE SUN, AND SHUDAN TIAN  \nABSTRACT. Hybridizable staggered discontinuous Galerkin methods are developed for arbitrary-order polyharmonic equations (−∆)mu = f on shape-regular polytopal meshes in Rd, for any m ≥ 1, d ≥ 2, and polynomial degree k ≥ 0. The method uses the mixed variable σ = ∇mu and a staggered primal–dual mesh to impose complementary continuity on scalar and tensor unknowns, without restrictions such as d ≥ m. Local trace and bubble enrichments stabilize low-order tensor spaces without adding global unknowns.  \nHybridization localizes the tensor variable and yields an equivalent stabilization-free weak Galerkin formulation. Well-posedness and optimal energy error estimates are proved, and numerical experiments on polygonal and tetrahedral meshes confirm the predicted rates.  \n1. INTRODUCTION  \nLet Ω be a bounded polyhedral domain in Rd , d ≥ 2. In this paper, we develop hybridizable staggered discontinuous Galerkin (HSDG) methods for the 2m-th order polyharmonic problem, for m ≥ 1,  \n(1)  \n(u( )∆uu···= , ∂mn−1u = 0 ,  \nin Ω , on ∂Ω .  \nThe method is designed for general polytopal meshes, arbitrary order m ≥ 1, arbitrary dimension d ≥ 2, and all polynomial degrees k ≥ 0. Its main ingredients are a staggered primal–dual mesh, a mixed tensor formulation, local tensor enrichments for loworder spaces, and a hybridization procedure that reduces the global system to scalar and face unknowns.  \nThe starting point is the mixed formulation  \n(2) σ = ∇mu, (−1)m divm σ = f. Formally, the relation σ = ∇mu can be written as  \n(σ,τ) = (−1)m (divm τ, u),  \nso the scalar unknown only needs to be in L2 (Ω), while the tensor unknown carries the H(divm )-type regularity. For m = 1, this is the classical mixed formulation based on H(div)-conforming spaces, including Raviart–Thomas, Brezzi–Douglas–Marini, Ndlec, and related elements [31, 8, 7, 30, 29] . For m = 2, several H(div div)-conforming finite element spaces have recently been developed [18, 14, 15, 11, 24] . For general m, constructing explicit H(divm )-conforming tensor finite elements is difficult, especially on general polytopes.  \n2020 Mathematics Subject Classification. 65N30; 65N12; 15A72 .  \nThe first author was partially supported by NSF grant DMS-2309785, the third author was supported by NSFC grant 12431014, and the fourth author was supported by NSFC grant 12401483 .  \n2  \nThe staggered discontinuous Galerkin (SDG) framework provides a different way to impose the mixed regularity. SDG methods, initiated by Chung and Engquist [20, 21], use staggered primal and dual grids to obtain stable and locally conservative discretizations. They have been extended from triangular meshes to polygonal meshes in [38, 39, 40], anda primal SDG method on polytopal meshes was recently developed in [19] . The present work extends [19] from second-order problems to arbitrary-order polyharmonic equations.  \nGiven a polyhedral primal mesh Kh , each element K is split by connecting an interior point of K to its faces, giving a refined mesh KRh . The dual mesh K∗h is formed by grouping refined cells that share a primal face. The key SDG idea is to impose complementary continuity on the primal and dual meshes. The tensor variable σh satisfies the required H(divm )-type conformity on dual elements, while the scalar variable uh is locally Hm-conforming on primal elements. Thus both variables are globally discontinuous, but their continuity properties are staggered and complementary. This gives a stable mixed discretization of (2) without constructing globally H(divm )-conforming tensor elements on the primal polytopal mesh.  \nThe scalar space is chosen elementwise as Pk+m(K) on each primal element. The main difficulty is to construct tensor spaces Σh,k for an arbitrary number of face-normal directions, order m, dimension d, and p","cbCailAJvlZWHNl7","https://ap.wps.com/l/cbCailAJvlZWHNl7","pdf",669762,5,1,28,"English","en",105,"# Introduction\n## Problem statement and mixed formulation\n## Staggered SDG framework and staggered regularity\n## Tensor space enrichments and stability\n## Hybridization and equivalent weak Galerkin formulation\n## Related work and context","[{\"question\":\"What kind of equations does the proposed HSDG method solve?\",\"answer\":\"It targets arbitrary-order polyharmonic equations (−∆)^m u = f on shape-regular polytopal meshes.\"},{\"question\":\"How does the method handle continuity requirements for the unknowns?\",\"answer\":\"It uses a staggered primal–dual mesh so the tensor variable satisfies H(div^m)-type conformity on dual elements, while the scalar variable is locally H^m-conforming on primal elements, yielding complementary staggered continuity.\"},{\"question\":\"Why are local trace and bubble enrichments needed?\",\"answer\":\"They stabilize low-order tensor spaces by recovering missing normal trace layers when k \\u003c m−1 and by providing enough tensor directions for local stability construction.\"},{\"question\":\"What results and evidence are provided to validate the method?\",\"answer\":\"The paper proves well-posedness and optimal energy error estimates, and numerical experiments on polygonal and tetrahedral meshes confirm the theoretical convergence rates.\"}]",1784197007,71,{"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":91,"head_meta":93,"extra_data":95,"updated_unix":28},"hybridizable-staggered-discontinuous-galerkin-methods-for-polyharmonic-equations-on-polytopes","",{"@graph":36,"@context":90},[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/hybridizable-staggered-discontinuous-galerkin-methods-for-polyharmonic-equations-on-polytopes/84598/",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","2026-07-22","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82,86],{"name":73,"@type":74,"acceptedAnswer":75},"What kind of equations does the proposed HSDG method solve?","Question",{"text":76,"@type":77},"It targets arbitrary-order polyharmonic equations (−∆)^m u = f on shape-regular polytopal meshes.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the method handle continuity requirements for the unknowns?",{"text":81,"@type":77},"It uses a staggered primal–dual mesh so the tensor variable satisfies H(div^m)-type conformity on dual elements, while the scalar variable is locally H^m-conforming on primal elements, yielding complementary staggered continuity.",{"name":83,"@type":74,"acceptedAnswer":84},"Why are local trace and bubble enrichments needed?",{"text":85,"@type":77},"They stabilize low-order tensor spaces by recovering missing normal trace layers when k \u003C m−1 and by providing enough tensor directions for local stability construction.",{"name":87,"@type":74,"acceptedAnswer":88},"What results and evidence are provided to validate the method?",{"text":89,"@type":77},"The paper proves well-posedness and optimal energy error estimates, and numerical experiments on polygonal and tetrahedral meshes confirm the theoretical convergence rates.","https://schema.org",{"og:url":52,"og:type":92,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":94,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":97},[98,102,106,110,114,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Story & 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