Solving steady interface problems using residual velocities
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Bibliographic record
Abstract
We consider elliptic problems in which the domain is separated into two regions by a steady free boundary, on which mixed Dirichlet–Neumann conditions are specified. Led by the classical Stefan condition applied to change of phase models, we consider numerical methods which evolve interfaces to the desired steady shape by using the residual in one of the boundary conditions as a normal velocity. Using linear perturbation analysis of simple cases, we show exactly which interfacial conditions lead to well-posed problems and which choices of velocities lead to convergent methods. Moreover, some velocities lead to methods having superior numerical properties, an idea related to early work of Garabedian. Analysis of a discrete scheme in which the free boundary is approximated by a cubic spline fit is presented, followed by an example computation.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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