Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms for\n Parabolic Problems
Bibliographic record
Abstract
We present a waveform relaxation version of the Dirichlet-Neumann and\nNeumann-Neumann methods for parabolic problems. Like the Dirichlet-Neumann\nmethod for steady problems, the method is based on a non-overlapping spatial\ndomain decomposition, and the iteration involves subdomain solves with\nDirichlet boundary conditions followed by subdomain solves with Neumann\nboundary conditions. For the Neumann-Neumann method, one step of the method\nconsists of solving the subdomain problems using Dirichlet interface\nconditions, followed by a correction step involving Neumann interface\nconditions. However, each subdomain problem is now in space and time, and the\ninterface conditions are also time-dependent. Using Laplace transforms, we show\nfor the heat equation that when we consider finite time intervals, the\nDirichlet-Neumann and Neumann-Neumann methods converge superlinearly for an\noptimal choice of the relaxation parameter, similar to the case of Schwarz\nwaveform relaxation algorithms. The convergence rate depends on the size of the\nsubdomains as well as the length of the time window. For any other choice of\nthe relaxation parameter, convergence is only linear. We illustrate our results\nwith numerical experiments.\n
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".