Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation
Bibliographic record
Abstract
Abstract Several numerical methods (orthogonal collocation, Galerkin, tau, least‐squares and least‐squares with a direct minimization algorithm) are applied to solve a linear diffusion–reaction problem. The spectral, finite‐element and spectral‐element frameworks are employed to investigate the methods. Overall, the Galerkin and tau methods are considered the most universal methods. Spectral framework: With sufficient diffusion limitations, the least‐squares method suffers in general from significantly lower accuracy than the Galerkin, tau and orthogonal collocation methods. On the other hand, the least‐squares method with a direct minimization algorithm provides favourable lower system matrix condition numbers than the conventional least‐squares approach. Hence, for higher diffusion limitations, the least‐squares direct minimization formulation provides higher numerical accuracy than the conventional least‐squares method, but is still not as accurate as the Galerkin, tau and orthogonal collocation techniques. The accuracy of the least‐squares solution can compete with the other methods only in cases with low gradients in the solution. Element framework: For a highly diffusion limited problem, the element framework is considered favourable as compared to the spectral framework. On the other hand, the element approach is not as efficient as the spectral solution for small Thiele modulus solutions.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".