Mixed finite element approximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements
Bibliographic record
Abstract
Mixed finite element methods are applied to a Poisson problem with singular-ity at a boundary point. The approximation spaces are based on quarter-point elements , the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarter-point elements at the coarse level, and subdividing them by mapping uniformly refined square meshes on the master element by their corresponding geometric transforms. For eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. For an improved version, using collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. The assembly of the required H(div)-conforming approximation spaces based on these kind of meshes are described. Results for a typical test problem demonstrate a superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hp-adaptive refinements. Specially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes.
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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 | 0.002 | 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 | 0.001 | 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".