Anisotropic space-time adaptation for reaction-diffusion problems
Bibliographic record
Abstract
A residual error estimator is proposed for the energy norm of the error for a scalar reaction-diffusion problem and for the monodomain model used in cardiac electrophysiology. The problem is discretized using $P_1$ finite elements in space, and the backward difference formula of second order (BDF2) in time. The estimator for space makes use of anisotropic interpolation estimates, assuming only minimal regularity. Reliability of the estimator is proven under certain mild assumptions on the convergence of the approximate solution. The monodomain model couples a nonlinear parabolic partial differential equation (PDE) with an ordinary differential equation (ODE) and this setting presents challenges theoretically as well as numerically. A space-time adaptation algorithm is proposed to control the global error, using a non-Euclidean metric for mesh adaptation and a simple method to adjust the time step. Numerical examples are used to verify the reliability and efficiency of the estimator, and to test the adaptive algorithm. The potential gains in efficiency of the proposed algorithm compared to methods using uniform meshes is discussed.
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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 | 0.000 | 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) | 0.000 | 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".