[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82858-en":3,"doc-seo-82858-105":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},82858,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Pressure-robust hp-a posteriori error estimates of H(div)-conforming discontinuous Galerkin methods for the Stokes equations","Develops and analyzes a pressure-robust residual-based hp-a posteriori error estimator for H(div)-conforming discontinuous Galerkin (dG) discretizations of the Stokes system on two- and three-dimensional polytopal Lipschitz domains. The estimator delivers an upper bound and a local lower bound for the velocity error in the energy norm, independent of pressure and robust with respect to viscosity. Error decomposition into conforming and nonconforming parts guides the proof, using partition-of-unity and local Helmholtz decompositions plus Bogovskiĭ-operator-based conforming estimators.","arXiv :2607 .04787v1 [math .NA] 6 Jul 2026  \nPressure-robust hp-a posteriori error estimates of H(div)-conforming discontinuous Galerkin methods for  \nthe Stokes equations  \nZhaonan Dong∗, Zuodong Wang†, Lina Zhao‡  \nJuly 7, 2026  \nAbstract  \nWe devise and analyze a pressure-robust residual-based hp-a posteriori error estimator for H(div)-conforming discontinuous Galerkin (dG) methods for the Stokes problem on two-and three-dimensional polytopal Lipschitz domains. The estimator provides an upper bound and a local lower bound for the velocity error in the energy norm, both robust with respect to the viscosity and independent of the pressure. Our analysis relies on a decomposition of the error into conforming and nonconforming parts. The nonconforming error is bounded using a partition-of-unity framework combined with local Helmholtz decompositions on vertex patches. The conforming error is analyzed by means of the generalized Bogovski˘ı operator of [14] in both two and three dimensions, yielding two pressure-independent residual-based estimators associated with different interpolation operators. In the first approach, the upper bound for the conforming error consists of five error indicators and a data oscillation term. Four of these indicators exhibit p-optimal scaling, while the remaining one is suboptimal by a factor of p 1/2 . In the second approach, the upper bound involves only two residual indicators together with the data oscillation term, at the expense of losing one order in p. Moreover, a pressure-robust local lower bound is established using H2-bubble functions inspired by techniques developed for fourth-order PDEs. Numerical results in two and three dimensions confirm the reliability, efficiency, and pressure-robustness of the proposed estimators.  \n1 Introduction  \nStokes problems arise in many scientific and engineering applications, including incompressible fluid dynamics, porous media flow, microfluidics, biological transport, and fluid–structure interaction. Let Ω be a polygonal or polyhedral domain in Rd , d = 2, 3, with Lipschitz boundary ∂Ω . The Stokes equations are given by  \n−µ∆u + ∇p = f in Ω ,∇·u = 0 in Ω ,  \nu = 0 on ∂Ω,  \nwhere the condition RΩ p dx = 0 is imposed to ensure the uniqueness of the pressure and, consequently, the unique solvability of the problem.  \nAmong the many discretisation techniques for the Stokes equations, pressure-robust methods have attracted considerable attention in recent years. A discretisation is said to be pressure-robust if the velocity error is independent of the pressure (or, equivalently, of irrotational forcing terms) . Classical mixed finite element methods, although satisfying the discrete inf–sup condition, typically yield velocity error estimates containing a pressure contribution scaled by µ −1 . As highlighted in the comprehensive review [29], this pressure dependence can significantly deteriorate the accuracy of the velocity approximation when the pressure is large or exhibits complex behaviour, even for simple benchmark problems such as the no-flow and stationary  \n1 Inria, 48 rue Barrault, 75647 Paris, France and CERMICS, CNRS, ENPC, Institut Polytechnique de Paris, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vall´ee, France. ([zhaonan.dong@inria.fr](zhaonan.dong@inria.fr))  \n2 School of Mathematical Sciences; Eastern Institute of Technology, Ningbo, Zhejiang, 315200, China. ([zdwang@eitech.edu.cn](zdwang@eitech.edu.cn)) .  \n3 Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China. ([linazha@cityu.edu.hk](linazha@cityu.edu.hk)) .  \nvortex cases. Pressure-robust methods overcome this limitation by enforcing a stronger form of the incompressibility constraint, either through exactly divergence-free velocity spaces (e.g., Scott–Vogelius elements and H(div)-conforming methods) or through suitable reconstructions of the test functions. These methods offer several important advantages, including velocity error estimates that are i","cbCaiiTXq9nv8usO","https://ap.wps.com/l/cbCaiiTXq9nv8usO","pdf",3979208,1,27,"English","en",105,"# Abstract\n# Introduction\n## Stokes problems and pressure dependence\n## Pressure-robust discretizations\n## Background on a posteriori error estimation","[{\"question\":\"What kind of error estimator is proposed for the Stokes problem?\",\"answer\":\"A pressure-robust residual-based hp-a posteriori error estimator for H(div)-conforming discontinuous Galerkin methods, providing upper and local lower bounds for the velocity error.\"},{\"question\":\"How does the estimator handle pressure dependence?\",\"answer\":\"Both the analysis and resulting bounds are constructed so the velocity error estimates are independent of the pressure (and thus of irrotational forcing effects), while remaining robust with respect to viscosity.\"},{\"question\":\"What mathematical techniques are used to bound the conforming and nonconforming parts of the error?\",\"answer\":\"The nonconforming part uses a partition-of-unity framework with local Helmholtz decompositions on vertex patches, while the conforming part is analyzed using a generalized Bogovskiĭ operator with different interpolation-based residual estimators.\"}]",1784183500,68,{"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},"pressure-robust-hp-a-posteriori-error-estimates-of-hdiv-conforming-discontinuous-galerkin-methods-for-the-stokes-equations","",{"@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/pressure-robust-hp-a-posteriori-error-estimates-of-hdiv-conforming-discontinuous-galerkin-methods-for-the-stokes-equations/82858/",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 kind of error estimator is proposed for the Stokes problem?","Question",{"text":75,"@type":76},"A pressure-robust residual-based hp-a posteriori error estimator for H(div)-conforming discontinuous Galerkin methods, providing upper and local lower bounds for the velocity error.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the estimator handle pressure dependence?",{"text":80,"@type":76},"Both the analysis and resulting bounds are constructed so the velocity error estimates are independent of the pressure (and thus of irrotational forcing effects), while remaining robust with respect to viscosity.",{"name":82,"@type":73,"acceptedAnswer":83},"What mathematical techniques are used to bound the conforming and nonconforming parts of the error?",{"text":84,"@type":76},"The nonconforming part uses a partition-of-unity framework with local Helmholtz decompositions on vertex patches, while the conforming part is analyzed using a generalized Bogovskiĭ operator with different interpolation-based residual estimators.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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