On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains
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Bibliographic record
Abstract
Abstract This study presents a semi-analytical approach to solve the Laplace equation in arbitrarily shaped two-dimensional domains. The method is meshless and addresses boundary value problems that include both pure Dirichlet and mixed Dirichlet-Neumann boundary conditions. The solution is obtained via a weighted superposition of harmonic polynomials which are obtained via an orthonormalization approach. We show that the approach is efficient in terms of number of operations. The numerical solution is convergent and exact (within machine precision) given a sufficient number of terms in the series. Moreover, the method offers several advantages over traditional approaches. Advantages include providing analytical expressions for the stream function and velocity components when solving potential flow problems. There are also important implications for the storage of model results; the method offers extremely low cost data storage. In the paper, several example applications involving arbitrary domains are presented. The results obtained are compared with known analytical solutions.
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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 | 0.005 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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