A Posteriori Error Estimates of Residual Type for Second Order Quasi-Linear Elliptic PDEs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator:\begin{equation*}\label{pde}\begin{array}{rcll}-\nabla \cdot (\alpha(x,\nabla u)\nabla u)&=& f(x) ~~~~& \mbox{in}~\Omega\subset\mathbb{R}^2, \\u&=& 0 &\mbox{on}~\partial\Omega,\end{array}\end{equation*}where $\Omega$ is assumed to be a polygonal bounded domain in $\mathbb{R}^2$, $f \in L^2(\Omega)$, and $\alpha$ is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the $H^1$-norm by an indicator $\eta$ which is composed of $L^2$- norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the $\alpha$-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.005 |
| 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