{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-10-30-afem-optimal-convergence-any-bulk-parameter-diening-storn-chatgpt/","md":"https://postcutoff.com/e/2026-10-30-afem-optimal-convergence-any-bulk-parameter-diening-storn-chatgpt/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-30-afem-optimal-convergence-any-bulk-parameter-diening-storn-chatgpt","date":"2026-10-30","date_precision":"day","short_title":"AFEM rate-optimal for any bulk parameter θ < 1 (ChatGPT framework)","deck":null,"takeaway":"Rate-optimality of the standard adaptive finite element method (Stevenson 2007; Cascón–Kreuzer–Nochetto–Siebert 2008) has always required the Dörfler marking parameter θ to be small. Lars Diening and Johannes Storn claim it for every θ in (0,1), for conforming Lagrange elements and the Poisson problem (arXiv 2610.39719). They say an extended discussion between Storn and ChatGPT 'led to an initial framework for the proof', and that they do not try to separate the human and AI contributions. The authors then verified and rewrote it. Not refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.39719: Optimal AFEM convergence for any bulk parameter smaller than one (Diening, Storn)","url":"https://arxiv.org/abs/2610.39719","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"PDF with the AI statement","url":"https://arxiv.org/pdf/2610.39719","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"],"title":"Adaptive finite elements: Diening and Storn claim rate-optimal AFEM convergence for every Dörfler bulk parameter θ < 1, with an initial proof framework from a ChatGPT discussion","summary":"arXiv 2610.39719 (\"Optimal AFEM convergence for any bulk parameter smaller than one\", Lars Diening and Johannes Storn) proves rate-optimal convergence of the standard adaptive finite element method with conforming Lagrange elements for the Poisson problem under Dörfler marking with any bulk parameter θ ∈ (0,1), removing the smallness restriction of the classical theory (Stevenson; Cascón, Kreuzer, Nochetto, Siebert; the 'axioms of adaptivity'). The proof framework emerged from a long interactive ChatGPT session.","key_facts":["Claim (abstract): 'We prove rate-optimal convergence of the standard adaptive finite element method with conforming Lagrange elements for the Poisson problem for Dörfler marking with any bulk parameter θ∈(0,1). This removes the smallness restriction on the bulk parameter that has persisted in the classical theory of optimal adaptive finite element methods.'","AI statement (verbatim): 'During the final stage of the present work, an extended interactive discussion between Johannes Storn and OpenAI's ChatGPT, motivated in particular by our recent results on L2 and Ritz projection stability, led to an initial framework for the proof. Since the mathematical ideas developed through a long sequence of questions, suggestions, and revisions, we do not attempt to separate the individual contributions of the human and AI participants to this stage. The authors subsequently verified, corrected, generalized, and substantially rewrote the arguments and take full responsibility for all content in this paper.'","Background (paper, §1): rate-optimality with coarsening by Binev–Dahmen–DeVore (2004), without coarsening by Stevenson (2007), with minimal refinement by Cascón–Kreuzer–Nochetto–Siebert (2008), organised into the 'axioms of adaptivity' by Carstensen–Feischl–Page–Praetorius (2014); all need θ small, although practitioners use larger θ","Authors: Lars Diening co-authored the 2008 Diening–Kreuzer contraction result cited as a milestone in the paper's history","Verification status: arXiv preprint, not refereed, no formal proof; no reaction found on 31 Oct"],"key_numbers":[],"tags":["math","numerical-analysis","finite-elements","adaptivity","chatgpt","preprint"],"science":{"field":"mathematics","subfield":"numerical analysis / adaptive finite element methods","problem":"Rate-optimality of AFEM with Dörfler marking for bulk parameters θ that are not small","result":"Claimed rate-optimal convergence for every θ ∈ (0,1) (Poisson problem, conforming Lagrange elements)","open_since":"2007","ai_system":["ChatGPT"],"human_role":"AI-assisted: an interactive ChatGPT discussion with Storn produced the initial proof framework; the authors verified, corrected, generalised and rewrote it","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint by specialists of the field"},"status":"pending","shock":""},"body_md":"## What happened\n\nAdaptive finite element methods refine a mesh only where an error estimator says the error is large. Dörfler marking refines a set of\nelements that carries a fraction θ of the estimated error. Since 2007 proofs that the method converges at the best possible rate have\nneeded θ to be small, even though larger θ is common in practice.\n\nOn 29 Oct Lars Diening and Johannes Storn posted a proof for every θ below 1. They write that an extended discussion between Storn and\nChatGPT gave the initial framework of the proof, and that the human and AI contributions at that stage cannot be separated.\n\n## Why it matters\n\nIt removes a long-standing technical restriction in the theory of adaptive methods. The paper also gives a clear first-hand account of\nmixed human–AI proof development. It is a preprint.","disputed":[{"label":"Verification","text":"Unrefereed preprint by specialists of the field","sources":[]}],"related":[{"id":"2026-10-31-ai-assisted-results-october-31-listing","url":"https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/","date":"2026-10-31","date_precision":"day","short_title":"AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct","deck":null,"takeaway":"The 31 Oct sweep listed 344 arXiv papers that mention AI, 75 conjecture papers with no AI statement and 48 uncited Hexagon works. We skimmed every AI statement and read about 70 PDFs or Hexagon records. Seven results got their own entries. Two come from senior authors: Panchenko's Mézard–Parisi formula ('found by ChatGPT Astra') and Bhargava's positive-density cyclicity of elliptic curves, written with GPT-6. The other five are Tianchi Yang's count of C₄-free graphs, Kolpakov's sharp dimension bounds for right-angled hyperbolic polytopes, the generalized Lax conjecture (Kummer–Netzer's AI-found generic case and OpenAI's counterexample), Diening–Storn's AFEM optimality for every bulk parameter, and an unverified GPT-6 Astra claim on Onsager's hard rods. The rest are listed here, among them Pongrácz's R(5,6) ≤ 84 certificate with GPT-6 Astra and Bowler–Pitz's negative answers to Halin-type questions, where Claude found key lemmas.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":42,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic","Google","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-31","type":"filed","text":"Created (science agent; arXiv abstract, introduction and AI statement read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-31","run":null,"sources_read":"Science agent; arXiv abstract, introduction and AI statement read","updated":"2026-10-31","human_review":null,"version":{"date":"2026-10-31"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":183,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":122,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":213,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":152,"in_training_data":false}],"short_url":null}