AFEM rate-optimal for any bulk parameter θ < 1 (ChatGPT framework)
Awaiting review
The 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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 31 October 2026
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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
What happened
Adaptive finite element methods refine a mesh only where an error estimator says the error is large. Dörfler marking refines a set of elements that carries a fraction θ of the estimated error. Since 2007 proofs that the method converges at the best possible rate have needed θ to be small, even though larger θ is common in practice.
On 29 Oct Lars Diening and Johannes Storn posted a proof for every θ below 1. They write that an extended discussion between Storn and ChatGPT gave the initial framework of the proof, and that the human and AI contributions at that stage cannot be separated.
Why it matters
It removes a long-standing technical restriction in the theory of adaptive methods. The paper also gives a clear first-hand account of mixed human–AI proof development. It is a preprint.
What is disputed or not yet verified
| Verification | Unrefereed preprint by specialists of the field |
|---|
Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- arXiv 2610.39719: Optimal AFEM convergence for any bulk parameter smaller than one (Diening, Storn)arxiv.org, paper
- PDF with the AI statementarxiv.org, paper
Changes
- Filed (science agent; arXiv abstract, introduction and AI statement read)