{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/h-1-stability-of-the-l-2-projection-onto","title":"$H^1$-Stability of the $L^2$-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes","arxiv_id":"2008.12759","date":"2020-08-28","proceeding":null,"authors":["Mazen Ali","Stefan A. Funken","Anja Schmidt"],"abstract":"The $L^2$-orthogonal projection $\\Pi_h:L^2(\\Omega)\\rightarrow\\mathbb{V}_h$ onto a finite element (FE) space $\\mathbb{V}_h$ is called $H^1$-stable iff $\\|\\nabla\\Pi_h u\\|_{L^2(\\Omega)}\\leq C\\|u\\|_{H^1(\\Omega)}$, for any $u\\in H^1(\\Omega)$ with a positive constant $C\\neq C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria for the $H^1$-stability of adaptively refined meshes. We show that adaptive refinement strategies for quadrilateral meshes in 2D (Q-RG and Q-RB), introduced originally in Bank et al. 1982 and Kobbelt 1996, are $H^1$-stable for FE spaces of polynomial degree $p=2,\\ldots,9$.","url_abs":"https://arxiv.org/abs/2008.12759v2","url_pdf":"https://arxiv.org/pdf/2008.12759v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"h-1-stability-of-the-l-2-projection-onto","repo_url":"https://github.com/aschmidtuulm/h1-stability","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}