Global convergence and local superconvergence of first-kind Volterra integral equation approximations
Bibliographic record
Abstract
We present a comprehensive convergence analysis for discontinuous piecewise polynomial approximations of a first-kind Volterra integral equation with smooth convolution kernel, examining the attainable order of (super-) convergence in collocation, quadrature discontinuous Galerkin (QDG) and full discontinuous Galerkin (DG) methods. We introduce new polynomial basis functions with properties that greatly simplify the convergence analysis for collocation methods. This also enables us to determine explicit formulae for the location of superconvergence points (i.e., discrete points at which the convergence order is one higher than the global bound) for all convergent collocation schemes. We show that a QDG method, which is based on piecewise polynomials of degree m and uses exactly m + 1 quadrature points and nonzero quadrature weights, is equivalent to a collocation scheme, and so its convergence properties are fully determined by the previous collocation analysis and they depend only on the quadrature point location (in particular, they are completely independent of the accuracy of the quadrature rule). We also give a complete analysis for QDG with more than m + 1 quadrature points when the degree of precision (d.o.p.) is at least 2m + 1. The behaviour (but not the approximation) is the same as that for a DG scheme when the d.o.p. is at least 2m + 2. Numerical test results confirm that the theoretical convergence rates are optimal.
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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 | 0.005 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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 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".