Achieving h‐ and p‐Robust Monolithic Multigrid Solvers for the Stokes Equations
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Bibliographic record
Abstract
ABSTRACT The numerical analysis of higher order mixed finite‐element discretizations for saddle‐point problems, such as the Stokes equations, has been well‐studied in recent years. While the theory and practice of such discretizations is now well‐understood, the same cannot be said for efficient preconditioners for solving the resulting linear (or linearized) systems of equations. In this work, we propose and study variants of the well‐known Vanka relaxation scheme that lead to effective geometric multigrid preconditioners for both the conforming Taylor‐Hood discretizations and nonconforming ‐ discretizations of the Stokes equations. Numerical results demonstrate robust performance with respect to FGMRES iteration counts for increasing polynomial order for some of the considered discretizations and expose open questions about stopping tolerances for effectively preconditioned iterations at high polynomial order.
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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 | 0.001 |
| 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 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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