Unique continuation principles for finite-element discretizations of the Laplacian
Bibliographic record
Abstract
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations are a natural tool to help gain understanding into elliptic equations, it is natural to ask if such principles also hold at the discrete level. In this work, we prove a version of the unique continuation principle for piecewise-linear and -bilinear finite-element discretizations of the Laplacian eigenvalue problem on polygonal domains in R 2 . Namely, we show that any solution to the discretized equation − Δ u = λ u with vanishing Dirichlet and Neumann traces must be identically zero under certain geometric and topological assumptions on the resulting triangulation. We also provide a counterexample, showing that a nonzero inner solution exists when the topological assumptions are not satisfied. Finally, we give an application to an eigenvalue interlacing problem, where the space of inner solutions makes an explicit appearance.
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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".