CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Joint fill-distance and kernel-width refinement for value and derivative consistency

Statement

Assume the hypotheses of Explicit cellwise error for kernel value and gradient quadrature and let f∈C³(R^d) have bounded derivatives. There are constants C₀,C₁, independent of ε,h and the interior point x, such that

$∣ Q_{ε, h}^{0} f (x) - f (x) ∣ \leq C_{0} ε^{2} ∥ D^{2} f ∥_{\infty} + C_{1} h (∥ \nabla f ∥_{\infty} + ε^{- 1} ∥ f ∥_{\infty}),$

$∣ Q_{ε, h}^{1} f (x) - \nabla f (x) ∣ \leq C_{0} ε^{2} ∥ D^{3} f ∥_{\infty} + C_{1} h (ε^{- 1} ∥ \nabla f ∥_{\infty} + ε^{- 2} ∥ f ∥_{\infty}) .$

Thus ε→0 and h/ε²→0 suffice for both value and derivative errors to vanish uniformly at interior points. Constants depend only on the fixed kernel, dimension, and cell regularity.

Facts & Assumptions

Given: An admissible kernel, a cell rule with the fill-radius and local-volume bounds of Meshfree cell quadrature and fill radius, h≤ε≤1, an interior point, and f∈C³ with bounded f and derivatives through order three.

[F1]

Second-order convolution bias for an interior C² field: |S_εf−f|≤(m₂/2)ε²||D²f||∞.

[F2]

Second-order bias for the kernel particle gradient: |D_εf−∇f|≤(m₃/6)ε²||D³f||∞.

[F3]

Explicit cellwise error for kernel value and gradient quadrature: the value and derivative quadrature errors are bounded by the two displayed h-terms.

Proof

1.1F1F3algebra

Add and subtract S_εf(x): |Q⁰_{ε,h}f−f|≤|Q⁰_{ε,h}f−S_εf|+|S_εf−f|. Apply [F1] and the first bound in [F3] to obtain the first inequality after increasing constants.

1.2F2F3algebra

Add and subtract D_εf(x): |Q¹_{ε,h}f−∇f|≤|Q¹_{ε,h}f−D_εf|+|D_εf−∇f|. Apply [F2] and the second bound in [F3] to obtain the second inequality.

2.1step 1.1step 1.2givenalgebra∎

If ε→0 and h/ε²→0, then ε²→0, h/ε²→0, and h/ε≤h/ε² because ε≤1. Every term on the right sides of steps 1.1 and 1.2 therefore tends to zero uniformly at the stated interior points.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)