Optimal L~∞ analysis of non-singular finite element methods/finite volume methods for the stationary 3D Navier-Stokes equations
Bibliographic record
Abstract
In this paper,we develop and analyze the L~∞ stability and convergence analysis for non-singular finite element and finite volume solutions for the stationary 3D Navier-Stokes equations.We obtain optimal estimates for the gradient of velocity and the pressure in the L~∞-norm by applying the stabilization of a macro-element and technical lemmas including weighted L~2-norm estimates for the regularized Green's functions associated with the Stokes problem.Moreover,using the finite element solutions as interpolations,the relationship between the finite element method and the finite volume method is used to obtain the interesting super-close convergence rate with O(h~(3/2)) in the L~2-norm and the optimal rate with 0(h) in the L~∞-norm between the finite element method and the finite volume method for the velocity gradient and the pressure.Furthermore,optimal error estimates in the L~∞-norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor O(|logh|) for the stationary 3D Naiver-Stokes equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".