[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84230-en":3,"doc-seo-84230-105":28,"detail-sidebar-cat-0-en-105":90},{"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":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},84230,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","An Approximate Bounded Cochain Projection","The paper constructs a projector from an infinite-dimensional Hilbert complex of differential k-forms onto a finite-dimensional piecewise polynomial sub-complex. On contractable domains the projector satisfies all Bounded Cochain Projector requirements: idempotence, uniform boundedness in the Sobolev norm, and exact commutation with the exterior derivative. On non-contractible domains idempotence and uniform boundedness persist, while commutation holds to arbitrary accuracy, enabling controlled approximation in de Rham sequence settings.","arXiv :2607 .07457v1 [math .NA] 8 Jul 2026  \nAN APPROXIMATE BOUNDED COCHAIN PROJECTION  \nMARC GERRITSMA∗ AND SUYASH SHRESTHA†  \nAbstract. This paper presents a construction of a projector from an infinite-dimensional Hilbert complex of differential k-forms onto a finite-dimensional piecewise polynomial sub-complex. We demonstrate that, on contractable domains, the proposed projector attains the three properties of the Bounded Cochain Projector, namely that the projector is idempotent, uniformly bounded in the Sobolev norm, and it commutes with the exterior derivative. On non-contractible domains, the projector remains idempotent and uniformly bounded, while the commuting property is satisfied up to arbitrary accuracy.  \nKey words. Bounded Cochain Projection, de Rham sequence.  \nAMS subject classifications. 35A35, 46N40, 47A46  \n1. Introduction. Let Ω be an n-dimensional manifold with sufficiently smooth Lipschitz boundary. We consider the closed Hilbert complex of differential forms introduced in [1],  \nR ,→ HΛ(0) (Ω)  HΛ(1) (Ω)  · · ·  HΛ(n−1)(Ω)  HΛ(n)(Ω) −→ 0.  \nThe exactness of this sequence depends on the global topology of Ω . In particular, if Ω is contractible, the de Rham complex is exact, meaning that the range of the exterior derivative equals the null space of this operator in the subsequent space. In contrast, when Ω is not contractible, the complex possesses a nontrivial cohomology, and the range of the exterior derivative forms only a subset of the null space of the subsequent exterior derivative.  \nFor each space HΛ(k)(Ω), we introduce a finite-dimensional subspace HΛh(k)(Ω) ⊂ HΛ(k)(Ω) , chosen such that the exterior derivative maps between successive finite-dimensional function spaces. This yields a finite-dimensional subcomplex  \nR ,→ HΛ(0)h(Ω)  HΛ(1)h(Ω)  · · ·  HΛh(n−1)(Ω)  HΛh(n)(Ω) −→ 0.  \nA bounded cochain projection, Π (k), maps elements of HΛ(k)(Ω) onto HΛh(k)(Ω) such that the projector is idempotent, i.e. Π (k)Π (k) = Π (k), the projected solution is bounded in the Sobolev norm, ∥Π(k)ϕ (k) ∥HΛ(k)(Ω) ≤ C∥ϕ(k) ∥HΛ(k)(Ω) for all ϕ (k) ∈ HΛ(k)(Ω), and the projector commutes with the exterior derivative, dΠ (k) = Π (k+1)d leading to the following commuting diagram  \n...  HΛ(k−1)(Ω)  \nΠ(k−1)  \n...  HΛh(k−1)(Ω)  \nHΛ(k)(Ω)  \nd  \nHΛ(k+1)(Ω)  \n...  \nΠ(k) Π(k+1)  \nHΛh(k)(Ω)  d  HΛh(k+1)(Ω)  ...  \nIn [1, 2] the bounded cochain projection was introduced. The construction of local bounded cochain projections is presented in [10] and is extended to double complexes in [9] . Subsequent work, [3, 6], refined these ideas in several directions, including local constructions, improved approximation properties, preservation of boundary conditions, and extensions to general finite element systems. More recently, [8] developed local L2-bounded commuting projections by replacing the continuous regularisation used in previous constructions, such as [7], with local discrete problems posedon an auxiliary refinement. Their construction establishes fully local commuting projections with provable local L2 stability while avoiding the analytical difficulties associated with continuous right inverses of the exterior derivative. This represents the first completely discrete construction of locally L2-bounded commuting projections. Likewise, in [5], commuting L2-projections on non-matching interfaces are presented. These papers employ finite-dimensional function spaces using Finite Element Exterior Calculus (FEEC) . Similar results are obtained with the Mimetic Spectral Element Method, [11] . The only requirement on the finite-dimensional subspaces is that they form a de Rham sequence, i.e. dHΛh(k)(Ω) ⊂ HΛh(k+1)(Ω) .  \nThe goal of the present work is to formulate a construction of a projector that fulfils the three aforementioned properties of a bounded cochian projection. The paper is organised as follows. In Section 2 some notation is introduced followed by the introduction of the canonical projector in Section 3. In Section 4 a mix","cbCaicmrL6McDOvx","https://ap.wps.com/l/cbCaicmrL6McDOvx","pdf",327591,1,"English","en",105,"# Introduction\n## Hilbert complex and de Rham exactness\n## Finite-dimensional subcomplex and bounded cochain projector\n# Prerequisites\n## Norms and inner products\n## Closed, exact, and harmonic forms","[{\"question\":\"What does the constructed projector map between?\",\"answer\":\"It maps differential k-forms from an infinite-dimensional Hilbert complex onto a finite-dimensional piecewise polynomial sub-complex.\"},{\"question\":\"Which properties hold on contractable domains?\",\"answer\":\"The projector is idempotent, uniformly bounded in the Sobolev norm, and commutes exactly with the exterior derivative.\"},{\"question\":\"What changes on non-contractible domains?\",\"answer\":\"Idempotence and uniform boundedness remain, but the commutation with the exterior derivative is achieved only up to arbitrary accuracy.\"}]",1784194198,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":26},"an-approximate-bounded-cochain-projection","",{"@graph":34,"@context":84},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/an-approximate-bounded-cochain-projection/84230/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the constructed projector map between?","Question",{"text":74,"@type":75},"It maps differential k-forms from an infinite-dimensional Hilbert complex onto a finite-dimensional piecewise polynomial sub-complex.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Which properties hold on contractable domains?",{"text":79,"@type":75},"The projector is idempotent, uniformly bounded in the Sobolev norm, and commutes exactly with the exterior derivative.",{"name":81,"@type":72,"acceptedAnswer":82},"What changes on non-contractible domains?",{"text":83,"@type":75},"Idempotence and uniform boundedness remain, but the commutation with the exterior derivative is achieved only up to arbitrary accuracy.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":44,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":44,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":44,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":44,"category_name":124,"show_sort_weight":27,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":27,"doc_module":4,"doc_module_name":44,"category_name":127,"show_sort_weight":27,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":44,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":44,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]