Accurate approximate solution of partial differential equations at off-mesh points
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Bibliographic record
Abstract
Numerical methods for partial differential equations often determine approximations that are more accurate at the set of discrete meshpoints than they are at the “off-mesh” points in the domain of interest. These methods are generally most effective if they are allowed to adjust the location of the mesh points to match the local behavior of the solution. Different methods will typically generate their respective approximations on incompatible, unstructured meshes, and it can be difficult to evaluate the quality of a particular solution, or to visualize important properties of a solution. In this paper we will introduce a generic approach which can be used to generate approximate solution values at arbitrary points in the domain of interest for any method that determines approximations to the solution and low-order derivatives at meshpoints. This approach is based on associating a set of “collocation” points with each mesh element and requiring that the local approximation interpolate the meshpoint data and almost satisfy the partial differential equation at the collocation points. The accuracy associated with this interpolation/collocation approach is consistent with the “meshpoint accuracy” of the underlying method. The approach that we develop applies to a large class of methods and problems. It uses local information only and is therefore particularly suitable for implementation in a parallel or network computing environment. Numerical examples are given for some second-order problems in two and three dimensions.
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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 | 0.000 | 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.006 | 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