Adaptive numerical solution of eigenvalue problems arising from finite element models. AMLS vs. AFEM
Bibliographic record
Abstract
We discuss adaptive numerical methods for the solution of eigenvalue problems arising either from the finite element discretization of a partial differential equation (PDE) or from discrete finite element modeling. When a model is described by a partial differential equation, the adaptive finite element method starts from a coarse finite element mesh which, based on a posteriori error estimators, is adaptively refined to obtain eigenvalue/eigenfunction approximations of prescribed accuracy. This method is well established for classes of elliptic PDEs, but is still in its infancy for more complicated PDE models. For complex technical systems, the typical approach is to directly derive finite element models that are discrete in space and are combined with macroscopic models to describe certain phenomena like damping or friction. In this case one typically starts with a fine uniform mesh and computes eigenvalues and eigenfunctions using projection methods from numerical linear algebra that are often combined with the algebraic multilevel substructuring method to achieve an adequate performance. These methods work well in practice but their convergence and error analysis is rather difficult. We analyze the relationship between these two extreme approaches. Both approaches have their pros and cons which are discussed in detail. Our observations are demonstrated with several numerical examples.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".