Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations
Bibliographic record
Abstract
We establish the well-posedness of an hp-version time-stepping discontinuous Galerkin method for the numerical solution of fractional superdiffusion evolution problems. In particular, we prove the existence and uniqueness of approximate solutions for generic hp-version finite element spaces featuring nonuniform time steps and variable approximation degrees. We then derive new hp-version error estimates in a nonstandard norm, which are completely explicit in the local discretization and regularity parameters. As a consequence, we show that by employing geometrically refined time steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are achieved for solutions with singular (temporal) behaviour near t=0 caused by the weakly singular kernel. Moreover, we show optimal algebraic convergence rates for h-version approximations on graded meshes. We present a series of numerical tests where we verify experimentally that our theoretical convergence properties also hold true in the stronger L∞ norm.
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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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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 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".