LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Second-order spatial consistency of the periodic FV heat stencil

Statement

Let $u_{r} \in C^{4} (T_{L}^{d})$ , and let $P_{h} u_{r}$ denote its samples on the periodic rectangular grid of Periodic rectangular reaction-diffusion grid. For every grid node $i$ ,

∣ (A_{h}^{p e r} P_{h} u_{r})_{i} + P_{h} (Δ u_{r})_{i} ∣ \leq \frac{1}{12} \sum_{j = 1}^{d} h_{j}^{2} ∥ \partial_{j j j j} u_{r} ∥_{L^{\infty} (T_{L}^{d})} .

For $D = diag (d_{1}, \dots, d_{m})$ , define the sampled spatial residual $r_{h} = D (P_{h} Δ u - Δ_{h} P_{h} u)$ , where $Δ_{h} = - A_{h}^{p e r}$ acts componentwise. If the indicated fourth derivatives are uniformly bounded for $0 \leq t \leq T$ , then $∥ r_{h} (t) ∥_{\infty} \leq C_{s p} h_{\max}^{}$ with $C_{s p} = \frac{d_{\max}}{12} \sum_{j} ∥ \partial_{j j j j} u ∥_{L^{\infty}}$ componentwise, uniformly in the mesh.

Facts & Assumptions

Given: A smooth periodic scalar or vector field on $T_{L}^{d}$ , the periodic rectangular grid, and its positive stencil $A_{h}^{p e r}$ .

[F1]

Periodic rectangular reaction-diffusion grid: $(A_{h}^{p e r} v)_{i} = \sum_{j} (2 v_{i} - v_{i + e_{j}} - v_{i - e_{j}}) / h_{j}^{2}$ with periodic wraparound.

[F2]

The Lagrange and Cauchy forms of Taylor's remainder: for a four-times continuously differentiable scalar function, the third-degree Taylor remainder at $x \pm h$ has magnitude at most $∥ f^{(4)} ∥_{\infty} h^{4} / 24$ .

[A1]

Each component $u_{r}$ is periodic and belongs to $C^{4}$ ; the derivative norms in the Statement are finite.

Proof

1.1A1F2algebra

Fix a node $i$ and direction $j$ . Lift the periodic component to its smooth periodic function on $R^{d}$ . Apply [F2] to the one-variable restrictions at $x_{i} + h_{j} e_{j}$ and $x_{i} - h_{j} e_{j}$ , expanding each through its cubic term at $x_{i}$ . The odd first- and third-degree terms cancel in the sum, leaving $u_{r} (x_{i} + h_{j} e_{j}) - 2 u_{r} (x_{i}) + u_{r} (x_{i} - h_{j} e_{j}) = h_{j}^{2} \partial_{j j} u_{r} (x_{i}) + ρ_{i j}$ with $∣ ρ_{i j} ∣ \leq h_{j}^{4} ∥ \partial_{j j j j} u_{r} ∥_{\infty} / 12$ .

2.1F1step 1.1algebra∎

Divide the identity of step 1.1 by $h_{j}^{2}$ , change sign to match the positive stencil, and sum over the finite directions $j = 1, \dots, d$ . The triangle inequality gives the first displayed bound. Multiplying componentwise by $d_{r}$ gives the residual estimate, and $h_{j} \leq h_{\max}$ gives the uniform constant stated above.

Depends on

Used by

Mesh-uniform first-order Lie error for a smooth periodic reaction-diffusion solution Theorem

Dependency tree · two levels

12 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

Willem Hundsdorfer, Numerical Solution of Advection-Diffusion-Reaction Equations, lecture notes for Ph.D. course, 1996 (standard reference, not scraped)