ExampleConstruction: AI-adaptedVerification: 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.

A periodic Q1/Q1 checkerboard pressure mode is invisible to divergence

Example

Take the periodic square with a $2 \times 2$ unit-square Q1 mesh. At the grid node $(i, j)$ , for $i, j \in {0, 1, 2}$ , set $q_{i j} = (- 1)^{i + j}$ and identify opposite boundaries periodically. Assume $τ_{K} > 0$ on every cell. These nodal values define a nonzero mean-zero Q1 pressure. It is invisible to the ordinary velocity-divergence coupling, although its PSPG pressure-gradient term is strictly positive.

Facts & Assumptions

Given: The periodic $2 \times 2$ Q1 mesh, equal-order Q1 velocity and pressure, strictly positive cell parameters $τ_{K}$ , and the PSPG residual term from Residual PSPG and grad-div terms for equal-order flow spaces.

[F1]

Residual PSPG and grad-div terms for equal-order flow spaces: PSPG adds $\sum_{K} τ_{K} (\nabla q_{h}, R_{M, K} (u_{h}, p_{h}))_{K}$ ; for the Stokes specialization and $u_{h} = 0$ , $p_{h} = q_{h}$ , $f = 0$ , the residual is $\nabla q_{h}$ .

Verification

1.1givenconstructalgebra

On a cell $K_{i j}$ with local coordinates $(ξ, η) \in [0, 1]^{2}$ , the corner values are $σ_{i j} (1, - 1, - 1, 1)$ with $σ_{i j} = (- 1)^{i + j}$ . Q1 interpolation therefore gives $q_{h} ∣_{K_{i j}} = σ_{i j} (1 - 2 ξ) (1 - 2 η)$ . Its integral on every cell is zero, so its global mean is zero, and it is nonzero at every mesh node.

2.1step 1.1algebra

For any Q1 velocity component $v_{1}$ , $\partial_{x} v_{1}$ on a cell is affine in $η$ and independent of $ξ$ . Since $\int_{0}^{1} (1 - 2 ξ) d ξ = 0$ , $\int_{K_{i j}} q_{h} \partial_{x} v_{1} = 0$ . Similarly, $\partial_{y} v_{2}$ is independent of $η$ and $\int_{0}^{1} (1 - 2 η) d η = 0$ , so $\int_{K_{i j}} q_{h} \partial_{y} v_{2} = 0$ . Summing cells gives $(q_{h}, \nabla \cdot v_{h}) = 0$ for every Q1 velocity.

3.1F1step 1.1step 2.1givenalgebra∎

On each unit cell, $\int_{K} ∣ \nabla q_{h} ∣^{2} = 4 / 3 + 4 / 3 = 8 / 3$ . Thus $\sum_{K} τ_{K} ∥ \nabla q_{h} ∥_{0, K}^{2} = (8 / 3) \sum_{K} τ_{K} > 0$ by the strict positivity in Given. By [F1], the PSPG term detects this mode, while the unstabilized divergence equation does not. This finite mode check proves neither a mesh-uniform inf-sup bound nor general PSPG accuracy.

Depends on

Residual PSPG and grad-div terms for equal-order flow spaces

Used by

Grad-div alone does not remove the Q1 checkerboard pressure kernel Example

Dependency tree · one level

1 result within one dependency step 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

Felipe Galarce and Douglas R. Q. Pacheco, Fully consistent lowest-order finite element methods for generalised Stokes flows with variable viscosity, arXiv:2411.02100v1 (standard reference, not scraped)
Thomas J. R. Hughes, Multiscale and Stabilized Methods (standard reference, not scraped)