Multilevel Methods for $p$-Adaptive Finite Element Analysis of Electromagnetic Scattering
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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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".