LOCAL CHEBYSHEV PROJECTION–INTERPOLATION OPERATOR AND APPLICATION TO THE h–p VERSION OF THE FINITE ELEMENT METHOD IN THREE DIMENSIONS
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Bibliographic record
Abstract
In this paper, we construct local Chebyshev projection–interpolation operators for tetrahedral and hexahedral elements in three dimensions based on the framework of the Jacobi-weighted Sobolev and Besov spaces. A simple assembly of the local Chebyshev projection–interpolations Π Ω j u on all the elements Ω j , 1 ≤ j ≤ J, leads to a globally continuous and piecewise polynomial which possesses the best approximation properties locally and globally. By applying the local Chebyshev projection–interpolation operators to the h–p version of the finite element method with general tetrahedral or hexahedral meshes for second-order elliptic problems in three dimensions, we establish the convergence rate for problems with homogeneous Dirichlet boundary conditions and the solution [Formula: see text], and for problems with nonhomogeneous Dirichlet boundary conditions and the solution [Formula: see text].
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it