Node-Diamond approximation of heterogeneous and anisotropic diffusion systems on arbitrary two-dimensional grids
Bibliographic record
Abstract
We develop a new nodal numerical scheme for solving diffusion equations. Anisotropic and heterogeneous diffusion tensors are taken into account in these equations. The method allows covering a wide range of general meshes such as non-confirming and distorted ones. The main idea consists in deriving the scheme from a discrete bilinear form using cellwise approximation of the diffusion tensor and particular discrete gradients. These gradients are conceived on diamonds partitioning the cell using local geometrical objects. The degrees of freedom are placed at the centers and vertices of cells. The cell unknowns can be eliminated without any fill-in. As a result, the coercivity of the scheme holds true unconditionally by construction. The convergence theorem of the Node-Diamond scheme is proved under classical assumptions on the physical parameters of the model equation and the mesh. Numerical results show the good behavior of the proposed approach on various examples among which we consider strongly anisotropic and heterogeneous systems. For instance, optimal accuracy consisting of quadratic rates for L2-errors and linear rates for H1-errors is obtained.
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How this classification was reachedexpand
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 | 0.001 |
| 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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".