Localized pointwise a posteriori error estimates for nonconforming finite element methods

[2608.02132] Localized pointwise a posteriori error estimates for nonconforming finite element methods Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2608.02132 (math) [Submitted on 3 Aug 2026 ( v1 ), last revised 23 Aug 2026 (this version, v2)] Title: Localized pointwise a posteriori error estimates for nonconforming finite element methods Authors: Yongxing Guo , Yuwen Li View PDF HTML (experimental) Abstract: This paper establishes the first localized pointwise a posteriori error estimates for nonconforming finite element discretizations of the Poisson and biharmonic equations. For the Poisson problem, we derive localized estimates for the function-value and broken gradient errors of the Crouzeix--Raviart method. For the Morley discretization of the biharmonic equation, we derive a localized a posteriori estimate that controls the local Hessian error. This provides the first pointwise a posteriori error analysis for the biharmonic equation, for either conforming or nonconforming finite element methods. The key ingredient, absent from pointwise analysis of second order PDEs, is the design of two novel weight functions that facilitate sharp estimates of the $L^1$ norms of derivatives of a regularized Green's function for the biharmonic operator. Comments: 22 pages, 4 figures Subjects: Numerical Analysis (math.NA) MSC classes: 65N15, 65N30 Cite as: arXiv:2608.02132 [math.NA] (or arXiv:2608.02132v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2608.02132 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yuwen Li [ view email ] [v1] Mon, 3 Aug 2026 12:21:12 UTC (190 KB) [v2] Sun, 23 Aug 2026 00:11:10 UTC (209 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-08 Change to browse by: cs cs.NA math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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