A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem
Bibliographic record
Abstract
In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier–Stokes equations and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernoulli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, to prove, under suitable small data assumptions, the existence of a unique solution to the continuous scheme, and the existence of solution to the associated Galerkin system for a feasible choice of the corresponding finite element subspaces. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".