[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83752-en":3,"doc-seo-83752-105":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},83752,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Virtual element methods for a class of fully nonlinear elliptic PDEs","Virtual element discretizations are developed for a variational formulation in H2 associated with Hamilton–Jacobi–Bellman and Isaacs equations under Cordes coefficients. Using H2-conforming virtual element spaces yields a streamlined analysis that avoids discrete Miranda–Talenti estimates typical of nonconforming schemes. The work studies how the polynomial degree of projection operators, especially for lower-order terms, impacts error analysis and robustness. It also presents a weak Dirichlet boundary imposition via Nitsche-type ideas and validates results through numerical experiments comparing scheme variants.","arXiv :2607 .03850v1 [math .NA] 4 Jul 2026  \nVirtual element methods for a class of fully nonlinear  \nelliptic PDEs  \nGuillaume Bonnet∗ Andrea Cangiani† Andreas Dedner‡  \nRicardo H. Nochetto§  \nAbstract  \nWe study virtual element discretizations of a well-known variational formulation in H2 of Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We show that the use of H2 conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda–Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.  \n1 Introduction  \nIn this work we are concerned with the efficient discretization of a general class of Isaacs fully nonlinear equations, which encompass the Hamilton–Jacobi–Bellman (HJB) equations as a special case. Fully nonlinear elliptic partial differential equations play a central role in several areas of applied mathematics, including optimal control, differential geometry, differential games, and image processing [27 , 33] . The numerical approximation of such problems remains challenging due to their nondivergence structure, strong nonlinearity, and the limited regularity typically available for their solutions [34, 20, 27, 36, 37, 28, 35, 39, 31, 29, 11, 16] .  \nWe propose and analyze arbitrary-order H 2-conforming Virtual Element Methods (VEM) for uniformly elliptic Isaacs equations. The approach builds upon our previous work on H 2-conforming VEM for linear elliptic problems in nondivergence form [12], thereby extending the methodology to the fully nonlinear setting, and accommodating for the presence of lower order terms.  \nMore specifically, we seek approximations of strong solutions of Isaacs equations satisfying the Cordes condition on convex domains. In this setting, well-posedness is a consequence of the Miranda–Talenti inequality, and is established through a variational formulation in H 2 (rather than H 1 ) . This has been used in the literature to design discontinuous Galerkin [36 , 37] and other, mostly nonconforming finite element methods [28, 35, 31, 39, 25, 29, 16] .  \n∗ CEREMADE, CNRS, Université Paris–Dauphine, Université PSL, 75016 Paris, France †Mathematics Area, International School for Advanced Studies (SISSA), 34136 Trieste, Italy ‡Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK  \n§ Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA  \nThe Cordes condition amounts to uniform ellipticity in dimension two, and is more restrictive in higher dimensions. For completeness, it may be observed that another family of numerical schemes for fully nonlinear elliptic equations are monotone schemes, see e.g. [11, 20 , 27 , 10] . Monotone schemes do not require the Cordes condition or even uniform ellipticity, and approximate viscosity solutions rather than strong solutions; however, their order is limited to two [34] . Accordingly, the two families of methods tend to be favored in different settings.  \nAs already observed in [28 , 12], in the Cordes setting, the use of H 2-conforming discrete spaces enables a straightforward discretization of second-order operators in strong form, since the continuous Miranda–Talenti inequality directly applies at the discrete level. By contrast, the analysis of nonconforming finite element discretizations requires a suitable discrete analogue of Miranda–Talenti which is typically achieved by the inclusion of appropriate stabilizing te","cbCaieHtPVnYCkKr","https://ap.wps.com/l/cbCaieHtPVnYCkKr","pdf",3930345,5,1,32,"English","en",105,"# Introduction\n## Main contributions\n## VEM for Isaacs equations\n## Monotonicity and error estimates\n## Weak imposition of boundary conditions","[{\"question\":\"What class of PDEs does the paper address?\",\"answer\":\"The paper targets fully nonlinear uniformly elliptic Isaacs equations, which include Hamilton–Jacobi–Bellman equations as a special case.\"},{\"question\":\"Why is using H2-conforming virtual element spaces important?\",\"answer\":\"H2 conformity enables a simpler analysis at the discrete level by leveraging the Miranda–Talenti inequality directly, avoiding discrete Miranda–Talenti estimates needed for nonconforming schemes.\"},{\"question\":\"How does the polynomial degree of projection operators affect the method?\",\"answer\":\"The polynomial degree of the projection operators, particularly for lower-order terms, influences both the error analysis and the robustness. In the nonlinear setting, not all combinations are admissible or favorable for all diffusion, advection, and reaction contributions.\"}]",1784190210,81,{"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},"virtual-element-methods-for-a-class-of-fully-nonlinear-elliptic-pdes","",{"@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/virtual-element-methods-for-a-class-of-fully-nonlinear-elliptic-pdes/83752/",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","2026-07-25","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 class of PDEs does the paper address?","Question",{"text":76,"@type":77},"The paper targets fully nonlinear uniformly elliptic Isaacs equations, which include Hamilton–Jacobi–Bellman equations as a special case.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is using H2-conforming virtual element spaces important?",{"text":81,"@type":77},"H2 conformity enables a simpler analysis at the discrete level by leveraging the Miranda–Talenti inequality directly, avoiding discrete Miranda–Talenti estimates needed for nonconforming schemes.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the polynomial degree of projection operators affect the method?",{"text":85,"@type":77},"The polynomial degree of the projection operators, particularly for lower-order terms, influences both the error analysis and the robustness. In the nonlinear setting, not all combinations are admissible or favorable for all diffusion, advection, and reaction contributions.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]