Multilevel Methods for $p$-Adaptive Finite Element Analysis of Electromagnetic Scattering
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Bibliographic record
Abstract
In p-adaptive finite element analysis, the large, sparse matrix that arises can be block structured according to the hierarchical level of the unknowns. A multilevel preconditioner for the matrix is a V-cycle that starts by applying Gauss-Seidel to the highest level, then the next level down, and so on. On the other side of the V, Gauss-Seidel is applied in the reverse order. At the bottom of the V is the lowest order system, which typically is solved exactly with a direct solver. However, for a complex geometry even the lowest order system may be too large for direct factorization. Here an alternative is proposed: to continue the V-cycle downwards, first into a set of auxiliary, node-based spaces, then through a series of progressively smaller matrices generated by an algebraic multigrid method. The smallest matrix is solved by factorization. The method is applied to p-adaptive analysis of a five-resonator iris filter, a split-ring resonator loaded waveguide, a “buckyball” metallic frame surrounding a conducting sphere, and a noncommensurate frequency selective surface. Tetrahedral elements up to fourth order are used. The largest matrix has over 12 million rows and 0.6 billion nonzero entries.
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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 it