[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82664-en":3,"doc-seo-82664-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82664,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Sobolev Stability of the L2 Projection on Hybrid Meshes","Establishes Lp- and W1,p-stability for the L2-projection onto mapped Lagrange finite elements defined on hybrid meshes formed by triangles and convex quadrilaterals produced by adaptive mesh refinement. For polynomial tensor degree K, results include W1,2-stability for all K ≥ 2 under Q-RG and Q-RB refinements. The work extends earlier bounds for Q-RG/Q-RB starting from parallelogram meshes and builds on a quadrilateral extension of techniques from prior literature.","arXiv :2607 .02362v1 [math .NA] 2 Jul 2026  \nSOBOLEV STABILITY OF THE L2-PROJECTION  \nON HYBRID MESHES  \nLARS DIENING, VIKTORIA LINGERT, AND TABEA TSCHERPEL  \nAbstract. We establish Lp- and W 1,p-stability of the L2-projection onto mapped Lagrange finite elements on hybrid meshes consisting of triangles and convex quadrilaterals arising from adaptive mesh refinement. If K is the (tensor product) degree of polynomials of the discretisation, then we show, in particular, W 1 ,2-stability for all K ≥ 2 for the Q-RG and Q-RB refinements. This extends results by Ali, Funken, and Schmidt [AFS22] which hold for the range  \n2 ≤ K ≤ 9 for initial meshes consisting of parallelograms. Our proof relies on an extension of the technique by Diening, Storn and Tscherpel in [DST21] to  \ngeneral convex quadrilaterals.  \n1. Introduction  \nStability of the L2-projection in norms other than ∥·∥L2 is a key element in the numerical analysis of finite element methods. For example, for the heat equation W 1 ,2-stability is equivalent to the inf-sup stability and quasi-optimality of Galerkin methods (see [TV16]) . For nonlinear evolution problems stability properties provide an important ingredient in the proof of error estimates that do not rely on a coupling of time and space discretisation parameters [BDSW21] . Furthermore, decay properties of the L2-projection are instrumental for results on hp-preconditioning [GS25] .  \nLet Π denote the L2-projection onto the Lagrange finite element space Vh of a certain polynomial degree K ≥ 1. We aim for stability in Lp and W 1,p for p ∈ [1 , ∞ ], i.e., we want to show that  \n∥Πu∥p ≲ ∥u∥p and ∥∇Πu∥p ≲ ∥∇u∥p , holds for any u ∈ Lp (Ω) or u ∈ W1,p(Ω) uniformly in the mesh size, respectively.  \nAs proved in [BX91] for quasi-uniform meshes stability in W 1 ,2 follows from inverse estimates and by using a Sobolev stable operator like the Scott–Zhang operator [SZ90] or the Cl´ement operator [Cl´e75] . However, both for non-quasiuniform meshes and for p  2 establishing stability results is a highly challenging task. For the analysis of adaptive finite element methods it is important that Sobolev stability holds for graded meshes arising from adaptive mesh refinement, uniformly in the refinement level. Furthermore, stability estimates for p  2 are crucial in the error analysis of nonlinear evolution equations [BDSW21] .  \nIn dimension d = 1 it is known that Sobolev stability cannot hold without conditions on the mesh grading [CT87; BY14] . In fact, in [BY14] a counterexample to W 1 ,2-stability is presented for the lowest order case when neighbouring intervals  \n2010 Mathematics Subject Classification. 65N30, 65N50, 65N12, 65M60 .  \nKey words and phrases. L2-projection, Lp-stability, Sobolev stability, quadrilateral meshes, adaptive mesh refinement, Lagrange elements.  \n2 L. DIENING, V. LINGERT, AND T. TSCHERPEL  \ndiffer sufficiently in size. In higher dimensions analogous counterexamples are unknown, but all available stability results are based on certain grading assumptions. Specifically, this means that the local mesh sizes of neighbouring elements are not allowed to differ too strongly.  \nFor simplicial meshes in dimension d ≥ 2 numerous stability results of the L 2-projection are available. Beginning with the work on mildly graded triangulationsin [CT87; Bom06; EJ95], stability results were subsequently established for loworder spaces on highly graded meshes, such as those generated by adaptive mesh refinement [Ste01; Ste02; BPS02; Car02; Car04] . More recently, stability was established for a bounded range of polynomial degrees and on meshes exhibiting realistic grading [BY14; GHS16; GHS19] for dimension d = 2 and in [BY14] for dimension d = 3 . The most recent results in [DST21; DST26] build on the techniques developed in [BY14] and establish mesh grading results for BisecMT [Mau95; Tra97], the generalisation of the newest vertex bisection, in any space dimension d ∈ N. Taken together, these works establish W 1 ,","cbCaimuEWywUoXlm","https://ap.wps.com/l/cbCaimuEWywUoXlm","pdf",615922,1,24,"English","en",105,"# Introduction\n## Problem and stability goal\n## Background and known limitations\n## Prior results on simplicial and quadrilateral meshes\n## Hybrid meshes and related work\n## Contribution of this paper","[{\"question\":\"What stability properties are proved for the L2-projection on hybrid meshes?\",\"answer\":\"The paper proves Lp-stability and W1,p-stability of the L2-projection onto mapped Lagrange finite element spaces defined on hybrid meshes with triangles and convex quadrilaterals.\"},{\"question\":\"How does the polynomial degree K affect the main W1,2 stability result?\",\"answer\":\"For the Q-RG and Q-RB refinements, W1,2-stability holds for all polynomial tensor degrees K ≥ 2.\"},{\"question\":\"What earlier results does this work extend, and what is the key methodological step?\",\"answer\":\"It extends stability results due to Ali, Funken, and Schmidt, previously covering a limited range of degrees for hybrid meshes. The proof relies on extending a technique from earlier work to general convex quadrilaterals.\"}]",1784182154,60,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"sobolev-stability-of-the-l2-projection-on-hybrid-meshes","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/sobolev-stability-of-the-l2-projection-on-hybrid-meshes/82664/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What stability properties are proved for the L2-projection on hybrid meshes?","Question",{"text":75,"@type":76},"The paper proves Lp-stability and W1,p-stability of the L2-projection onto mapped Lagrange finite element spaces defined on hybrid meshes with triangles and convex quadrilaterals.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the polynomial degree K affect the main W1,2 stability result?",{"text":80,"@type":76},"For the Q-RG and Q-RB refinements, W1,2-stability holds for all polynomial tensor degrees K ≥ 2.",{"name":82,"@type":73,"acceptedAnswer":83},"What earlier results does this work extend, and what is the key methodological step?",{"text":84,"@type":76},"It extends stability results due to Ali, Funken, and Schmidt, previously covering a limited range of degrees for hybrid meshes. 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