[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84333-en":3,"doc-seo-84333-105":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},84333,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","3D Virtual Element Method for Advection-Diffusion-Reaction Problems with Variable Coefficients","Proposes and analyzes a Continuous Interior Penalty (CIP) stabilized three-dimensional Virtual Element Method (VEM) for advection-diffusion-reaction equations on general polyhedral meshes. Builds a novel Oswald-type quasi-interpolant variant to replace prior 2D analyses that depend on global quasi-uniformity and constant physical parameters. Derives robust, uniform error estimates in the hyperbolic limit under realistic local quasi-uniformity and variable coefficients. Includes three-dimensional numerical experiments confirming convergence rates and the absence of non-physical oscillations.","arXiv :2607 .08237v1 [math .NA] 9 Jul 2026  \n3D Virtual Element Method for  \nAdvection-Diffusion-Reaction Problems with Variable Coefficients on Locally Quasi-Uniform Polytopes  \nM. Trezzi∗1  \n1 Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di  \nMilano-Bicocca, Via Roberto Cozzi 55-20125 Milano, Italy  \nJuly 10, 2026  \nAbstract  \nIn this paper, we propose and analyze a Continuous Interior Penalty (CIP) stabilized Virtual Element Method (VEM) for three-dimensional advection-diffusion-reaction equationson general polyhedral meshes. While CIP-VEM schemes have been recently explored in a twodimensional setting, their analysis heavily relies on global mesh quasi-uniformity and constant physical parameters. We overcome these limitations by introducing a novel three-dimensional variant of the Oswald-type quasi-interpolant. This allows us to establish robust, uniform error estimates in the hyperbolic limit under a realistic local quasi-uniformity assumption and variable coefficients. Finally, we provide a comprehensive set of three-dimensional numerical experiments to validate the theoretical convergence rates and demonstrate the absence of non-physical oscillations.  \n1 Introduction  \nOver the past decade, the Virtual Element Method (VEM) has emerged as a powerful tool for the numerical solution of partial differential equations (PDEs) on general polygonal/polytopal meshes [3, 4] . By extending the foundational concepts of traditional Finite Element Methods (FEM), VEM bypasses element-geometry constraints, allowing for elements with an arbitrary number of edges and faces, non-convex shapes, and hanging nodes. This geometric versatility is a cornerstone of VEM’s success, driving its application across a wide spectrum of engineering problems, as highlighted in recent literature [27, 2] .  \nThe objective of this work is the development of a robust polyhedral discretization for threedimensional advection-diffusion-reaction equations in the hyperbolic limit. This type of problem poses numerical challenges in the advection-dominated regime. In particular, standard Galerkin approximations typically fail, resulting in numerical solutions with severe spurious oscillations across element boundaries. To overcome this issue, the FEM community has established several stabilization frameworks, including Streamline Upwind Petrov-Galerkin (SUPG) [13], Local Projection Stabilization (LPS) [22], upwind Discontinuous Galerkin (DG) formulations [12], and Continuous Interior Penalty (CIP) methods [15, 19] . Some of these techniques has been extend to a VEM framework in [8, 9, 5, 23, 21, 26] .  \nAmong the aforementioned stabilization techniques, the CIP approach stands out as a “minimal stabilization” paradigm, as originally observed in [14] . In fact, it has been shown that the distance between the continuous convective derivative and the discrete space scales precisely with the magnitude of the gradient jumps across element interfaces.  \nThe CIP technique was recently extended to the VEM framework within a two-dimensional setting in [7, 21] . Although these formulations yield robust estimates in the hyperbolic limit, their  \n∗[manuel.trezzi@unimib.it](manuel.trezzi@unimib.it)  \nnumerical analysis strictly relies on the assumption of global mesh quasi-uniformity and constant physical parameters. The present work aims to overcome these major restrictions. By introducing a novel three-dimensional variant of the Oswald-type quasi-interpolant, we are able to relax the global constraint to a realistic local quasi-uniformity assumption, while simultaneously handling variable coefficients. The Oswald interpolant is necessary to gain control over the advective term within the energy norm of the problem. Alternatively, one could avoid employing the Oswald operator, but only at the cost of sacrificing the control of the advective derivative in the norm. Such a choice results in a weaker norm that fails to provide robust control over transport ","cbCaibOmm5rjwr8P","https://ap.wps.com/l/cbCaibOmm5rjwr8P","pdf",1745642,1,27,"English","en",105,"# Introduction\n## Problem objective and stabilization motivation\n## Continuous Interior Penalty in VEM\n# Problem Setting and Virtual Element Discretization\n## Model problem","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses steady three-dimensional advection-diffusion-reaction PDEs on general polyhedral meshes, focusing on the hyperbolic (advection-dominated) regime where standard Galerkin methods create spurious oscillations.\"},{\"question\":\"How does the method improve over prior CIP-VEM analyses?\",\"answer\":\"It introduces a new three-dimensional Oswald-type quasi-interpolant variant that relaxes the need for global mesh quasi-uniformity and supports variable coefficients under a realistic local quasi-uniformity assumption.\"},{\"question\":\"Why is the Oswald interpolant important in the analysis?\",\"answer\":\"It enables control of the advective term in the energy norm; avoiding it weakens the norm and fails to provide robust control of transport instabilities. It also requires weak boundary condition enforcement via a Nitsche-VEM approach.\"}]",1784194884,68,{"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},"3d-virtual-element-method-for-advection-diffusion-reaction-problems-with-variable-coefficients","",{"@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/3d-virtual-element-method-for-advection-diffusion-reaction-problems-with-variable-coefficients/84333/",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 problem does the paper address?","Question",{"text":75,"@type":76},"It addresses steady three-dimensional advection-diffusion-reaction PDEs on general polyhedral meshes, focusing on the hyperbolic (advection-dominated) regime where standard Galerkin methods create spurious oscillations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method improve over prior CIP-VEM analyses?",{"text":80,"@type":76},"It introduces a new three-dimensional Oswald-type quasi-interpolant variant that relaxes the need for global mesh quasi-uniformity and supports variable coefficients under a realistic local quasi-uniformity assumption.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the Oswald interpolant important in the analysis?",{"text":84,"@type":76},"It enables control of the advective term in the energy norm; avoiding it weakens the norm and fails to provide robust control of transport instabilities. 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