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.

✓ 11 results · all verified · 7 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Projection, pressure correction and saddle-point solvers

1 · Prerequisites

2 · Summary

Projection, pressure correction and saddle-point solvers

Periodic staggered-grid setting

The projection results here concern the unit torus $T^{d}$ , a uniform periodic grid with $N \geq 2$ , face-centred velocity arrays, cell-centred pressure arrays, and the mean-zero pressure gauge. The exact index and sign conventions are fixed in Periodic staggered-grid velocity and pressure spaces. In particular, $Δ_{h} = D_{h} G_{h}$ is negative semidefinite, $A_{h} = - Δ_{h}^{V}$ is positive semidefinite, and $D_{h}$ and $G_{h}$ are negative adjoints. The constant pressure mode is removed by working in $Q_{h}$ ; the pressure Laplacian is then invertible by Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound. These claims use periodicity and the declared staggered placement. Collocated central differences have a distinct checkerboard null mode, shown in projection-and-pressure-correction-for-incompressible-flow-examples.

Incremental correction and exact projection

Backward-Euler incremental pressure correction on a periodic staggered grid defines the backward Euler predictor, mean-zero pressure-increment Poisson solve, velocity correction, and incremental pressure update. The well-definedness proof Well-definedness of the periodic pressure-correction solves verifies the predictor and pressure solves and preservation of the pressure gauge. Each force input has zero discrete mean in every face component. Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility identifies the velocity update with the mass-orthogonal projector onto $\ker D_{h}$ , so the corrected velocity is exactly discretely divergence-free.

The step structure is summarized below. The arrows show the dependencies between the three algebraic operations; every box is defined in the algorithm item, and the diagram is not used as a proof.

\begin{tikzpicture}[node distance=2.0cm, >=stealth, every node/.style={draw, rounded corners, align=center, minimum width=3.4cm, minimum height=0.9cm}]
  \node (pred) {Velocity predictor\\solve for provisional velocity};
  \node (poisson) [right of=pred, xshift=2.5cm] {Pressure increment\\solve mean-zero Poisson equation};
  \node (correct) [right of=poisson, xshift=2.5cm] {Velocity correction\\project to zero divergence};
  \node (pupdate) [below of=poisson, yshift=-1.2cm] {Pressure update\\add the computed increment};
  \draw[->] (pred) -- (poisson);
  \draw[->] (poisson) -- (correct);
  \draw[->] (poisson.south) -- (pupdate.north);
\end{tikzpicture}

The pressure-increment box feeds two updates: the velocity correction and the pressure gauge update. The latter uses the same increment and does not alter the velocity projection identity. Modified velocity-pressure energy identity for periodic pressure correction derives an exact modified velocity-pressure energy identity for this periodic backward-Euler scheme. No wall-boundary version of that identity is asserted.

BDF2 and rotational variants

BDF2 standard and rotational incremental pressure-correction variants records the standard incremental and rotational BDF2 variants, with the factor $2 k / 3$ in the velocity correction and the corresponding rotational pressure update. The algorithms have explicit periodic gauge conventions and their linear solves are well-defined by Well-definedness of the periodic pressure-correction solves. No stability, second-order error rate, or pressure rate is attached to these variants on this page.

Smooth periodic velocity accuracy

For smooth periodic solutions of the unsteady Stokes equations, the local sampling lemma Second-order consistency of staggered periodic sampling proves second-order spatial consistency of the staggered differences. Under the explicit compatibility and smoothness assumptions in First-order velocity accuracy for smooth periodic Stokes data, the backward-Euler incremental method has the velocity estimate $O (h^{2} + k)$ over a fixed time interval. The result concerns velocity only and does not claim a pressure-error rate. Its domain is the torus; bounded-wall pressure and boundary-layer rates require separate proofs. A tangential no-slip defect for a Neumann pressure correction is exhibited on the B page.

Saddle systems and conditional preconditioning bounds

Symmetric saddle-point system, Schur complement, and block preconditioners fixes the real symmetric saddle matrix, positive Schur matrix $S = B A^{- 1} B^{T}$ , exact block-LU solve, pressure-mass surrogate $γ M_{Q}$ , and Cholesky-symmetric preconditioning convention. The exact solve is verified in Exact Schur solve for a full-row-rank symmetric saddle system. The generic spectral equivalence Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling requires the stated inf-sup, continuity, and velocity-stiffness constants; Spectrum of a symmetrically pressure-mass-preconditioned saddle system translates those bounds into eigenvalue intervals. These are spectral statements, not claims of an iteration count or generic speed-up.

For the bounded-domain MINI pair, the steady Stokes corollaries Steady MINI Stokes pressure-mass Schur bounds and MINI Stokes block-preconditioned spectral intervals use the actual Fortin, Korn, and inf-sup hypotheses with viscosity scaling shown explicitly. The pressure-mass equivalence is for the steady velocity stiffness; no shifted unsteady Schur estimate is inferred from it.

Scope and source trail

The periodic projection formulas and their boundary caveats follow the survey of Guermond, Minev and Shen and the NPTEL lecture on pressure-velocity coupling. Block factorization and pressure-Schur scaling follow the saddle point survey by Benzi, Golub and Liesen. The local statements on this page retain their own assumptions and complete proofs; cited material does not replace those arguments.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Periodic staggered-grid velocity and pressure spaces

Definition

Fix an integer dimension $d \geq 1$ and an integer resolution $N \geq 2$ , where $N \in N$ (The natural numbers $N$ (von Neumann)), and put $h = 1 / N$ . Real scalar and array operations use the field $R$ (The reals form a field). Let $J_{N} = {0, \dots, N - 1}^{d}$ ; each coordinate index addition below is reduced to the unique element of ${0, \dots, N - 1}$ congruent modulo $N$ . The unit torus $T^{d}$ means the unit cube with opposite faces identified, so fields and arrays are periodic across those faces. Write $1 = (1, \dots, 1)$ and let $e_{i}$ be the $i$ th standard coordinate vector. Cell $j$ has centre $x_{j} = h (j + \frac{1}{2} 1)$ . The pressure value $q_{j}$ is stored at that centre. Velocity component $i$ is stored at the right face of cell $j$ , with centre $x_{j} + \frac{h}{2} e_{i}$ .

Let ${\tilde{Q}}_{h}$ be all real scalar cell arrays and let $Q_{h} = {q \in {\tilde{Q}}_{h} : h^{d} \sum_{j} q_{j} = 0}$ . Let $V_{h}$ be the real vector face arrays $v = (v_{i, j})_{1 \leq i \leq d, j \in J_{N}}$ . Give them the mass inner products $(q, r)_{Q_{h}} = h^{d} \sum_{j \in J_{N}} q_{j} r_{j}, (v, w)_{M_{h}} = h^{d} \sum_{i = 1}^{d} \sum_{j \in J_{N}} v_{i, j} w_{i, j} .$ Define the pressure gradient $G_{h} : {\tilde{Q}}_{h} \to V_{h}$ and velocity divergence $D_{h} : V_{h} \to Q_{h}$ by $(G_{h} q)_{i, j} = \frac{q_{j + e_{i}} - q_{j}}{h}, (D_{h} v)_{j} = \sum_{i = 1}^{d} \frac{v_{i, j} - v_{i, j - e_{i}}}{h} .$ The compatible pressure Laplacian is $Δ_{h} = D_{h} G_{h}$ ; its restriction to $Q_{h}$ is negative semidefinite, and $G_{h}$ annihilates constants. Define the componentwise vector Laplacian by $(Δ_{h}^{V} v)_{i, j} = h^{- 2} \sum_{ℓ = 1}^{d} (v_{i, j + e_{ℓ}} - 2 v_{i, j} + v_{i, j - e_{ℓ}}), A_{h} = - Δ_{h}^{V},$ and the discrete velocity-gradient seminorm by $∥ G_{h} v ∥_{M_{h}}^{2} = h^{d - 2} \sum_{i = 1}^{d} \sum_{ℓ = 1}^{d} \sum_{j \in J_{N}} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} .$ The fixed cell/face placement is part of the definition; collocated central differences are a different discretization.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Symmetric saddle-point system, Schur complement, and block preconditioners

Definition

Let $n, m \geq 1$ , $V = M_{n \times 1} (R)$ , and $Q = M_{m \times 1} (R)$ . Let $A \in M_{n} (R)$ be symmetric positive definite, let $B \in M_{m \times n} (R)$ have full row rank (meaning $B^{T} q = 0$ implies $q = 0$ ), and let $M_{Q} \in M_{m} (R)$ be symmetric positive definite. The saddle system and its matrix are $K = (\begin{array}{cc} A & B^{T} \\ B & 0 \end{array}), A u + B^{T} p = f, B u = g .$ Define the positive pressure Schur matrix $S : = B A^{- 1} B^{T} .$ In the block-LU convention of Block LU factorisation and the associated Schur complement, the trailing Schur complement of $A$ in $K$ is $- S$ . The exact block solve first solves $A u_{0} = f$ , then $S p = B u_{0} - g$ , and finally $A u = f - B^{T} p$ ; the theorem Exact Schur solve for a full-row-rank symmetric saddle system ↗ verifies this procedure and well-definedness.

For a scaling $γ > 0$ , the pressure-mass block-diagonal preconditioner is $P_{γ} = diag (A, γ M_{Q}) .$ It is symmetric positive definite. If $P_{γ} = C C^{T}$ is its Cholesky factorization with $C$ lower triangular and positive on the diagonal, the symmetric preconditioned system is $C^{- 1} K C^{- T} y = C^{- 1} r$ , with the usual change of unknown. This is the symmetric-preconditioning convention of Left preconditioning, right preconditioning, and symmetric preconditioning of a linear system. No spectral bound, Krylov iteration count, complexity estimate, or generic speed improvement is part of this definition.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound

Statement

Use the spaces and operators of Periodic staggered-grid velocity and pressure spaces with $d \geq 1$ , $N \geq 2$ , and $h = 1 / N$ . Then $Δ_{h} = D_{h} G_{h}$ maps $Q_{h}$ bijectively to itself, the kernel of $Δ_{h}$ on all cell arrays consists exactly of the constants, and $D_{h} v$ has zero mean for every $v \in V_{h}$ . In addition, every $q \in Q_{h}$ satisfies the resolution-independent bound $∥ q ∥_{Q_{h}} \leq \sqrt{d} ∥ G_{h} q ∥_{M_{h}} .$

Facts & Assumptions

Given: A fixed periodic grid with $d \geq 1$ , $N \geq 2$ , its cell and face indexing, and the operators $G_{h}, D_{h}, Δ_{h}$ and mass pairings.

[F1]

The pressure and velocity arrays, inner products, and operators have the staggered periodic conventions in Periodic staggered-grid velocity and pressure spaces.

[F2]

For a linear map between finite-dimensional spaces, $\dim V = \dim \ker T + \dim im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F3]

The real coordinate pairing on a finite array space is an inner product (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F4]

Cauchy–Schwarz holds in every real inner-product space (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

Proof

1.1F1algebra

Reindexing the finite periodic sums gives, for every cell array $q$ and face array $v$ , $(G_{h} q, v)_{M_{h}} = h^{d} \sum_{i, j} \frac{(q_{j + e_{i}} - q_{j}) v_{i, j}}{h} = - h^{d} \sum_{j} q_{j} \sum_{i} \frac{v_{i, j} - v_{i, j - e_{i}}}{h} = - h^{d} \sum_{j} q_{j} (D_{h} v)_{j} .$ In particular, when $q \in Q_{h}$ , this is $(G_{h} q, v)_{M_{h}} = - (q, D_{h} v)_{Q_{h}}$ , the negative-adjoint identity in the stated mass pairings.

1.2F1F3F4givenalgebra

Let $q \in Q_{h}$ . Since $q$ has mean zero, $q_{x} = N^{- d} \sum_{y \in J_{N}} (q_{x} - q_{y}),$ and [F3]–[F4] give the finite-sum Cauchy--Schwarz inequality $∥ q ∥_{Q_{h}}^{2} \leq N^{- 2 d} \sum_{x, y \in J_{N}} ∣ q_{x} - q_{y} ∣^{2} .$ For each ordered pair $(x, y)$ choose the path that changes coordinates in the order $1, \dots, d$ , moving forward modulo $N$ in each coordinate. It has at most $d N$ directed edges. Cauchy--Schwarz along the path bounds $∣ q_{x} - q_{y} ∣^{2}$ by $d N$ times the sum of squared edge differences on it.

2.1F1step 1.1

For any cell array $q$ , applying the explicit weighted-sum identity of step 1.1 with $v = G_{h} q$ gives $h^{d} \sum_{j} q_{j} (Δ_{h} q)_{j} = - (G_{h} q, G_{h} q)_{M_{h}} \leq 0.$ If $Δ_{h} q = 0$ , then $G_{h} q = 0$ , so $q_{j + e_{i}} = q_{j}$ for every $i, j$ . Coordinate steps connect all cells of the periodic grid, hence $q$ is constant. Every constant has zero gradient, so the kernel on all cell arrays is exactly the constants.

3.1F1step 2.1algebra

For every $v$ , summing $D_{h} v$ over cells cancels each face value once with its opposite occurrence, so $D_{h} v$ has zero mean. The same is true of $Δ_{h} q = D_{h} (G_{h} q)$ . Therefore $Δ_{h}$ restricts to a linear map $Q_{h} \to Q_{h}$ . Its kernel there is zero by step 2.1, and its domain and codomain are the same finite-dimensional space.

4.1F2step 3.1algebra

Rank-nullity [F2] now gives $\dim im Δ_{h} = \dim Q_{h}$ ; since the image is a subspace of $Q_{h}$ , the restriction is onto as well as one-to-one. Thus each mean-zero right-hand side has a unique mean-zero pressure solution.

5.1F1step 1.2algebra∎

Fix one directed edge in coordinate $i$ . For each coordinate $r < i$ , its value along the path is $y_{r}$ ; to use the fixed edge, $y_{r}$ is fixed and $x_{r}$ has at most $N$ choices. For each $r > i$ , the path still has value $x_{r}$ ; then $x_{r}$ is fixed and $y_{r}$ has at most $N$ choices. In coordinate $i$ there are at most $N^{2}$ ordered start/end pairs. Hence this edge occurs in at most $N^{d + 1}$ paths. Summing the path bounds and dividing by $N^{2 d}$ yields $∥ q ∥_{Q_{h}}^{2} \leq d N^{2 - d} \sum_{i, j} ∣ q_{j + e_{i}} - q_{j} ∣^{2} = d ∥ G_{h} q ∥_{M_{h}}^{2} .$ Taking nonnegative square roots proves the stated inequality.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Second-order consistency of staggered periodic sampling

Statement

Use Periodic staggered-grid velocity and pressure spaces with $h = 1 / N$ . For a smooth periodic vector field $u \in C^{4} (T^{d}; R^{d})$ , define $R_{h} u$ by sampling component $i$ at each face centre $x_{j} + h e_{i} / 2$ . For a smooth periodic scalar $p \in C^{3} (T^{d})$ , define $R_{h} p$ by sampling at cell centres $x_{j}$ . Put $∥ r ∥_{{\tilde{Q}}_{h}} = (h^{d} \sum_{j} r_{j}^{2})^{1 / 2}$ for all cell arrays $r$ ; this agrees with $∥ r ∥_{Q_{h}}$ on $Q_{h}$ . Then uniformly for $N \geq 2$ , $∥ D_{h} R_{h} u - R_{h} (\nabla \cdot u) ∥_{{\tilde{Q}}_{h}} + ∥ A_{h} R_{h} u - R_{h} (- Δ u) ∥_{M_{h}} + ∥ G_{h} R_{h} p - R_{h} (\nabla p) ∥_{M_{h}} \leq C h^{2},$ where $C$ depends only on $d$ and the suprema of the displayed fields' derivatives through orders $4$ for $u$ and $3$ for $p$ . Here the sampling of a vector field is facewise componentwise. For any $g \in C^{2} (T^{d})$ with zero integral, the cell-centred or face-centred midpoint samples have discrete mean whose absolute value is at most $C_{d} h^{2} ∥ D^{2} g ∥_{\infty}$ .

Facts & Assumptions

Given: The unit torus, the staggered periodic grid and operators, and the bounded derivatives in the statement.

[F1]

Cell and face centres, discrete differences, and weighted mass norms are defined in Periodic staggered-grid velocity and pressure spaces.

[F2]

The unit torus is partitioned into the $N^{d}$ periodic grid cells of side $h = 1 / N$ , centred at cell sampling points. Smooth fields on $T^{d}$ are used through their smooth $1$ -periodic lifts to $R^{d}$ , including across periodic seams.

[F3]

If $g$ has continuous derivatives through order $m + 1$ , Taylor's theorem gives a pointwise remainder $g (x) - \sum_{j = 0}^{m} g^{(j)} (c) (x - c)^{j} / j! = g^{(m + 1)} (ξ) (x - c)^{m + 1} / (m + 1)!$ (Taylor's Lagrange remainder follows from repeated Rolle steps).

[F4]

Differentiating the line restriction $t \mapsto g (x + t z)$ uses the ordinary chain rule (Sum, product, quotient, and chain rules from difference quotients).

Proof

1.1F1F2F3givenalgebra

For $g \in C^{3} (T^{d})$ , Taylor's formula with Lagrange remainder at the midpoint and [F3] with $m = 2$ give $g (x + h e_{i} / 2) - g (x - h e_{i} / 2) = h \partial_{i} g (x) + r, ∣ r ∣ \leq h^{3} ∥ \partial_{i}^{3} g ∥_{\infty} / 24.$ The same midpoint formula, with the two cell centres surrounding face $x_{j} + h e_{i} / 2$ , gives $(G_{h} R_{h} p)_{i, j} = \partial_{i} p (x_{j} + h e_{i} / 2) + O (h^{2}),$ with the pointwise error bounded by $h^{2} ∥ \partial_{i}^{3} p ∥_{\infty} / 24$ .

1.2F1F2F3givenalgebra

For $g \in C^{4} (T^{d})$ , [F3] with $m = 3$ at $x$ gives $g (x + h e_{i}) - 2 g (x) + g (x - h e_{i}) = h^{2} \partial_{i}^{2} g (x) + r,$ where the two fourth-order Lagrange remainders have total absolute value at most $h^{4} ∥ \partial_{i}^{4} g ∥_{\infty} / 12$ . Apply this in every coordinate to $u_{i}$ at its face centre. It follows that $(A_{h} R_{h} u)_{i, j} = - (Δ u_{i}) (x_{j} + h e_{i} / 2) + O (h^{2})$ , with the error bounded by $h^{2} \sum_{ℓ} ∥ \partial_{ℓ}^{4} u_{i} ∥_{\infty} / 12$ .

2.1F1F2F3step 1.1algebra

Applying the midpoint formula to component $u_{i}$ at cell centre $x_{j}$ shows, by [F3] with $m = 2$ , $(D_{h} R_{h} u)_{j} = \sum_{i} \partial_{i} u_{i} (x_{j}) + r_{j}, ∣ r_{j} ∣ \leq \frac{h^{2}}{24} \sum_{i} ∥ \partial_{i}^{3} u_{i} ∥_{\infty} .$ Thus the discrete divergence differs pointwise from the sampled continuous divergence by $O (h^{2})$ .

3.1F1F2step 1.1step 2.1step 1.2algebra

Each pointwise error bound in steps 1.1–1.2 is uniform. The weighted mass of the grid contains $h^{d} N^{d} = 1$ total cell volume and there are $d$ velocity components. Therefore the corresponding mass norms are bounded by the same constants (or their finite $d$ -component sum) times $h^{2}$ . Combining the divergence, vector Laplacian and pressure gradient bounds proves the first claim.

4.1F1F2F3F4givenalgebra∎

On each cell centred at $x_{j}$ , fix $x$ and put $z = x - x_{j}$ and $ψ (t) = g (x_{j} + t z)$ for $0 \leq t \leq 1$ . By [F3] with $m = 1$ and [F4], $g (x) = g (x_{j}) + \nabla g (x_{j}) \cdot z + r_{j} (x)$ with $∣ r_{j} (x) ∣ \leq C_{d} ∥ D^{2} g ∥_{\infty} ∣ z ∣^{2}$ . The linear term integrates to zero on the centred cell, so the difference between $h^{d} g (x_{j})$ and $\int_{C_{j}} g$ is at most $C_{d} h^{d + 2} ∥ D^{2} g ∥_{\infty}$ . Summing the $N^{d}$ cells yields $∣ h^{d} \sum_{j} g (x_{j}) - \int_{T^{d}} g (x) d x ∣ \leq C_{d} h^{2} ∥ D^{2} g ∥_{\infty} .$ The face centres for any fixed component likewise centre a translated partition into cells of side $h$ , so the same estimate holds for their samples. If the integral vanishes, this gives the stated discrete-mean bound.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Exact Schur solve for a full-row-rank symmetric saddle system

Statement

Under the hypotheses of Symmetric saddle-point system, Schur complement, and block preconditioners, $S = B A^{- 1} B^{T}$ is positive definite and hence invertible. For every $f \in R^{n}$ and $g \in R^{m}$ , the unique solution of $(\begin{array}{cc} A & B^{T} \\ B & 0 \end{array}) (\begin{array}{c} u \\ p \end{array}) = (\begin{array}{c} f \\ g \end{array})$ is $p = S^{- 1} (B A^{- 1} f - g), u = A^{- 1} (f - B^{T} p) .$ The factorization used in this solve is the published block-LU factorization An invertible leading block yields block LU through its Schur complement.

Facts & Assumptions

Given: $A$ symmetric positive definite, $B$ full row rank, and the finite-dimensional symmetric saddle equations with right-hand side $(f, g)$ .

[F1]

The saddle matrix, pressure Schur matrix $S = B A^{- 1} B^{T}$ , and the two equations are defined in Symmetric saddle-point system, Schur complement, and block preconditioners.

[F2]

If a square block matrix has an invertible leading block, it factorizes through the Schur complement as $(\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}) = (\begin{array}{cc} I & 0 \\ A_{21} A_{11}^{- 1} & I \end{array}) (\begin{array}{cc} A_{11} & A_{12} \\ 0 & A_{22} - A_{21} A_{11}^{- 1} A_{12} \end{array})$ (An invertible leading block yields block LU through its Schur complement).

[F3]

For a linear map between finite-dimensional spaces, $\dim V = \dim \ker T + \dim im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

Proof

1.1givenF1F3algebra

Positive definiteness of $A$ makes its kernel zero: if $A x = 0$ , then $x^{T} A x = 0$ , so the positivity hypothesis forces $x = 0$ . Since $A$ is a square matrix on a finite-dimensional space, [F3] makes its image full-dimensional; thus $A$ is invertible. For nonzero $q \in R^{m}$ , full row rank gives $B^{T} q \neq 0$ . Put $y = A^{- 1} B^{T} q$ , which is nonzero. Then $q^{T} S q = (B^{T} q)^{T} A^{- 1} (B^{T} q) = (A y)^{T} y = y^{T} A y > 0.$ Therefore $S$ is symmetric positive definite.

2.1F1F2step 1.1algebra

Apply [F2] to $K$ with leading block $A$ , upper-right block $B^{T}$ , lower-left block $B$ , and lower-right block $0$ . Its trailing Schur complement is $0 - B A^{- 1} B^{T} = - S$ . Hence $K = (\begin{array}{cc} I & 0 \\ B A^{- 1} & I \end{array}) (\begin{array}{cc} A & B^{T} \\ 0 & - S \end{array}),$ and both diagonal blocks of the upper factor are invertible by the hypotheses and step 1.1.

3.1F1step 2.1algebra∎

Solve the lower-triangular factor against $(f, g)$ first. It gives the intermediate vector $(f, g - B A^{- 1} f)$ . The second block row of the upper solve is $- S p = g - B A^{- 1} f$ , so $p = S^{- 1} (B A^{- 1} f - g)$ . The first row gives $u = A^{- 1} (f - B^{T} p)$ . Substitution verifies $A u + B^{T} p = f$ and $B u = B A^{- 1} f - S p = g$ . Invertibility of the diagonal blocks in step 2.1 makes both triangular solves unique, so this is the unique solution.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling

Statement

Let $V, Q$ be finite-dimensional real inner-product spaces with norms $∥ \cdot ∥_{V}, ∥ \cdot ∥_{Q}$ . Let $a (v, w) = v^{T} A w$ be symmetric positive definite, and let $b (v, q) = q^{T} B v$ be a bilinear constraint form, represented by the matrices in Symmetric saddle-point system, Schur complement, and block preconditioners. Suppose that, for constants $α, Λ, C_{b}, β > 0$ , $α ∥ v ∥_{V}^{2} \leq a (v, v) \leq Λ ∥ v ∥_{V}^{2}, ∣ b (v, q) ∣ \leq C_{b} ∥ v ∥_{V} ∥ q ∥_{Q}, \sup_{v \neq 0} \frac{∣ b (v, q) ∣}{∥ v ∥_{V}} \geq β ∥ q ∥_{Q} .$ If $M_{Q}$ is the pressure mass matrix for $∥ \cdot ∥_{Q}$ , then the positive Schur matrix $S = B A^{- 1} B^{T}$ satisfies, for every $q \in Q$ , $\frac{β^{2}}{Λ} q^{T} M_{Q} q \leq q^{T} S q \leq \frac{C_{b}^{2}}{α} q^{T} M_{Q} q .$ Thus pressure-mass Schur equivalence is conditional on the actual inf-sup, continuity, and velocity stiffness scalings shown here.

Facts & Assumptions

Given: Finite-dimensional real spaces $V, Q$ , an SPD velocity matrix $A$ , a constraint matrix $B$ , a pressure mass matrix $M_{Q}$ , and the positive constants in the statement.

[F1]

The saddle matrix, its velocity and constraint blocks, and the pressure Schur matrix $S = B A^{- 1} B^{T}$ use the conventions of Symmetric saddle-point system, Schur complement, and block preconditioners.

[F2]

The positive-definite bilinear form $a$ is an inner product with norm $∥ v ∥_{A} = \sqrt{a (v, v)}$ (Real and complex inner product spaces, with the inner product linear in the first argument).

[F3]

Cauchy–Schwarz holds in the inner product $a$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

Proof

1.1F1F2F3algebra

Fix $q \in Q$ and put $w = A^{- 1} B^{T} q$ . By [F1] and symmetry of $A$ , $b (v, q) = q^{T} B v = v^{T} B^{T} q = v^{T} A w = a (v, w)$ for every $v \in V$ . By [F3], $∣ a (v, w) ∣ \leq ∥ v ∥_{A} ∥ w ∥_{A}$ . If $w \neq 0$ , equality is attained at $v = w$ ; if $w = 0$ , both sides are zero. Hence $\sup_{v \neq 0} \frac{∣ b (v, q) ∣}{\sqrt{a (v, v)}} = \sqrt{a (w, w)} = \sqrt{q^{T} B A^{- 1} B^{T} q} .$

2.1givenstep 1.1algebra

The upper energy inequality in the statement implies $\sqrt{a (v, v)} \leq \sqrt{Λ} ∥ v ∥_{V}$ , so the supremum in step 1.1 is at least $β ∥ q ∥_{Q} / \sqrt{Λ}$ . The lower energy inequality implies $\sqrt{a (v, v)} \geq \sqrt{α} ∥ v ∥_{V}$ ; continuity of $b$ therefore makes that supremum at most $C_{b} ∥ q ∥_{Q} / \sqrt{α}$ . Squaring these inequalities and using step 1.1 proves both displayed Schur bounds.

3.1F1givenstep 2.1algebra∎

For nonzero $q$ , the lower bound in step 2.1 is positive because $β > 0$ and $M_{Q}$ is positive definite. Thus $S$ is positive definite, and the two quadratic-form inequalities are the stated pressure-mass spectral equivalence.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Backward-Euler incremental pressure correction on a periodic staggered grid

Definition

Fix $d \geq 1$ , $N \geq 2$ , $h = 1 / N$ , viscosity $ν > 0$ , and time step $k > 0$ . Given $u^{n} \in V_{h}$ with $D_{h} u^{n} = 0$ , a mean-zero pressure $p^{n} \in Q_{h}$ , and a force $f^{n + 1} \in V_{h}$ whose components each have zero discrete mean, $h^{d} \sum_{j \in J_{N}} f_{i, j}^{n + 1} = 0 (1 \leq i \leq d),$ define the next step by the following two solves and update:

Find ${\tilde{u}}^{n + 1} \in V_{h}$ satisfying $\frac{{\tilde{u}}^{n + 1} - u^{n}}{k} + ν A_{h} {\tilde{u}}^{n + 1} + G_{h} p^{n} = f^{n + 1} .$
Find the unique $ϕ^{n + 1} \in Q_{h}$ satisfying $Δ_{h} ϕ^{n + 1} = k^{- 1} D_{h} {\tilde{u}}^{n + 1} .$
Set $u^{n + 1} = {\tilde{u}}^{n + 1} - k G_{h} ϕ^{n + 1}, p^{n + 1} = p^{n} + ϕ^{n + 1} .$

The pressure is kept in $Q_{h}$ after every solve; the constant pressure mode is set to zero. The procedure is only asserted for the periodic staggered grid and the admissible data stated here. The separate well-definedness theorem Well-definedness of the periodic pressure-correction solves ↗ verifies both solves and the preserved constraints.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

BDF2 standard and rotational incremental pressure-correction variants

Definition

Fix $d \geq 1$ , $N \geq 2$ , $h = 1 / N$ , $ν > 0$ , $k > 0$ , and a step $n \geq 1$ . Supply $u^{n - 1}, u^{n} \in V_{h}$ with zero discrete divergence, a mean-zero $p^{n} \in Q_{h}$ , and $f^{n + 1} \in V_{h}$ whose components each have zero discrete mean, $h^{d} \sum_{j \in J_{N}} f_{i, j}^{n + 1} = 0 (1 \leq i \leq d) .$ Compute a predictor ${\tilde{u}}^{n + 1} \in V_{h}$ from $\frac{3 {\tilde{u}}^{n + 1} - 4 u^{n} + u^{n - 1}}{2 k} + ν A_{h} {\tilde{u}}^{n + 1} + G_{h} p^{n} = f^{n + 1},$ then solve for the unique mean-zero $ϕ^{n + 1} \in Q_{h}$ from $Δ_{h} ϕ^{n + 1} = \frac{3}{2 k} D_{h} {\tilde{u}}^{n + 1} .$ Set $u^{n + 1} = {\tilde{u}}^{n + 1} - (2 k / 3) G_{h} ϕ^{n + 1}$ . Choose exactly one pressure update:

Standard incremental: $p^{n + 1} = p^{n} + ϕ^{n + 1}$ .
Rotational incremental: $p^{n + 1} = p^{n} + ϕ^{n + 1} - ν D_{h} {\tilde{u}}^{n + 1}$ .

In both branches the pressure is gauged in $Q_{h}$ . Initial values $u^{0}, u^{1}, p^{1}$ are supplied by the caller; no initialization accuracy, stability, or convergence rate is asserted here. The separate well-definedness theorem Well-definedness of the periodic pressure-correction solves ↗ verifies the predictor and pressure solves for either branch.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Spectrum of a symmetrically pressure-mass-preconditioned saddle system

Statement

Assume the hypotheses and Schur bounds of Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling, with constants $α, Λ, C_{b}, β > 0$ . For $γ > 0$ , use $P_{γ} = diag (A, γ M_{Q})$ from Symmetric saddle-point system, Schur complement, and block preconditioners, and set $a_{γ} = \frac{β^{2}}{Λ γ}, b_{γ} = \frac{C_{b}^{2}}{α γ}, λ_{\pm} (s) = \frac{1 \pm \sqrt{1 + 4 s}}{2} .$ Let $μ_{1}, \dots, μ_{m}$ be the eigenvalues, with multiplicity, of $(γ M_{Q})^{- 1 / 2} S (γ M_{Q})^{- 1 / 2}$ . Then $a_{γ} \leq μ_{j} \leq b_{γ}$ , and the symmetric square-root-preconditioned saddle matrix has eigenvalues ${λ_{-} (μ_{j}), λ_{+} (μ_{j}) : 1 \leq j \leq m} and 1 with multiplicity n - m .$ Consequently its spectrum is contained in $[λ_{-} (b_{γ}), λ_{-} (a_{γ})] \cup {1} \cup [λ_{+} (a_{γ}), λ_{+} (b_{γ})],$ where the eigenvalue $1$ has multiplicity zero if $n = m$ . This is a spectral estimate for the stated scaled block preconditioner, not an iteration-count or generic solver-speed claim.

Facts & Assumptions

Given: Real finite-dimensional spaces $V, Q$ , an SPD velocity matrix $A$ , full-row-rank constraint matrix $B$ , pressure mass matrix $M_{Q}$ , and the inf-sup, continuity, and energy-equivalence hypotheses of the Schur theorem.

[F1]

The saddle matrix is $K = (\begin{array}{cc} A & B^{T} \\ B & 0 \end{array})$ , the positive Schur matrix is $S = B A^{- 1} B^{T}$ , and $P_{γ} = diag (A, γ M_{Q})$ is the block preconditioner (Symmetric saddle-point system, Schur complement, and block preconditioners).

[F2]

The Schur quadratic form satisfies $(β^{2} / Λ) M_{Q} \leq S \leq (C_{b}^{2} / α) M_{Q}$ (Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling).

[F3]

Every self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F4]

Rank-nullity gives $\dim V = \dim \ker T + \dim im T$ for a linear map on a finite-dimensional space (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

Proof

1.1F1F2F3algebra

By [F3], the SPD matrices $A$ and $γ M_{Q}$ have symmetric positive square roots and inverse square roots. Put $T = (γ M_{Q})^{- 1 / 2} B A^{- 1 / 2}, X = P_{γ}^{- 1 / 2} K P_{γ}^{- 1 / 2} = (\begin{array}{cc} I & T^{T} \\ T & 0 \end{array}) .$ Then $T T^{T} = (γ M_{Q})^{- 1 / 2} S (γ M_{Q})^{- 1 / 2}$ . Congruencing [F2] by $(γ M_{Q})^{- 1 / 2}$ shows $a_{γ} I \leq T T^{T} \leq b_{γ} I$ . Hence $T T^{T}$ is positive definite and all its eigenvalues $μ_{j}$ lie in $[a_{γ}, b_{γ}]$ .

2.1F1F3step 1.1algebra

By [F3], choose an orthonormal eigenbasis $y_{1}, \dots, y_{m}$ of $T T^{T}$ , with eigenvalues $μ_{j} > 0$ . Set $x_{j} = T^{T} y_{j} / \sqrt{μ_{j}}$ . Then $T x_{j} = \sqrt{μ_{j}} y_{j}$ and $x_{i}^{T} x_{j} = \frac{y_{i}^{T} T T^{T} y_{j}}{\sqrt{μ_{i} μ_{j}}} = δ_{i j},$ so the $x_{j}$ are orthonormal. On the two-dimensional space spanned by $(x_{j}, 0)$ and $(0, y_{j})$ , $X$ has matrix $(\begin{array}{cc} 1 & \sqrt{μ_{j}} \\ \sqrt{μ_{j}} & 0 \end{array}),$ whose characteristic polynomial is $λ^{2} - λ - μ_{j}$ . Its two roots are $λ_{-} (μ_{j})$ and $λ_{+} (μ_{j})$ .

3.1F1F4step 1.1step 2.1algebra

The matrix $T T^{T}$ is positive definite by step 1.1, so for every $y \in R^{m}$ , $x = T^{T} (T T^{T})^{- 1} y$ satisfies $T x = y$ . Thus $T : R^{n} \to R^{m}$ is onto, and [F4] gives $\dim \ker T = n - m$ . For $x \in \ker T$ , $X (x, 0) = (x, 0)$ , giving eigenvalue $1$ on this kernel. The $m$ orthonormal pairs from step 2.1 and this $n - m$ dimensional kernel account for all $n + m$ dimensions and hence for the full spectrum of $X$ .

4.1F1F3step 1.1step 3.1algebra∎

If $C$ is the Cholesky factor in the symmetric-preconditioning definition, then $P_{γ} = C C^{T}$ . Let $O = P_{γ}^{- 1 / 2} C$ ; step 1.1 gives $O O^{T} = I$ , so $O$ is orthogonal and $C^{- 1} K C^{- T} = O^{T} (P_{γ}^{- 1 / 2} K P_{γ}^{- 1 / 2}) O .$ Therefore the published Cholesky-symmetric preconditioning has the same spectrum as $X$ . Since $λ_{-}$ decreases and $λ_{+}$ increases on $s > 0$ , the eigenvalue intervals follow from step 1.1.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Well-definedness of the periodic pressure-correction solves

Statement

On the periodic staggered grid of Periodic staggered-grid velocity and pressure spaces, every admissible backward- Euler step of Backward-Euler incremental pressure correction on a periodic staggered grid and every admissible BDF2 step of BDF2 standard and rotational incremental pressure-correction variants has a unique predictor, a unique mean-zero pressure increment, and a unique updated pair. Both algorithms preserve the zero-mean pressure gauge and exact discrete incompressibility.

Facts & Assumptions

Given: $d \geq 1$ , $N \geq 2$ , $h = 1 / N$ , $ν > 0$ , $k > 0$ , admissible mean-zero data, and either stated pressure-correction algorithm.

[F1]

In the backward-Euler algorithm the predictor matrix is $I + k ν A_{h}$ ; in BDF2 it is $(3 / (2 k)) I + ν A_{h}$ . The right-hand sides and pressure updates are exactly those in the two algorithm definitions (Backward-Euler incremental pressure correction on a periodic staggered grid, BDF2 standard and rotational incremental pressure-correction variants).

[F2]

The periodic componentwise Laplacian and the seminorm satisfy $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} \geq 0$ by finite periodic reindexing (Periodic staggered-grid velocity and pressure spaces).

[F3]

$D_{h} v$ has zero mean and $Δ_{h} : Q_{h} \to Q_{h}$ is bijective (Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound).

[F4]

A linear endomorphism of a finite-dimensional space with zero kernel is invertible by rank-nullity (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ); $V_{h}$ is finite-dimensional because it is a finite face-array space.

Proof

1.1givenF1F2F4algebra

For any nonzero $v \in V_{h}$ , the componentwise Laplacian definition and finite periodic reindexing give $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} \geq 0.$ Therefore $((I + k ν A_{h}) v, v)_{M_{h}} = ∥ v ∥_{M_{h}}^{2} + k ν (A_{h} v, v)_{M_{h}} > 0$ . The predictor matrix has zero kernel and is invertible by [F4], so the backward-Euler predictor has a unique solution.

2.1givenF1F2F4step 1.1algebra

Similarly, $(((3 / (2 k)) I + ν A_{h}) v, v)_{M_{h}} = (3 / (2 k)) ∥ v ∥_{M_{h}}^{2} + ν (A_{h} v, v)_{M_{h}} > 0$ . The BDF2 predictor matrix has zero kernel and is invertible by [F4], so its predictor is unique for either pressure-update branch.

3.1F1F3step 1.1step 2.1

For either predictor, summing $D_{h} \tilde{u}$ over all cells cancels the two occurrences of each periodic face value. Hence the right-hand side of the corresponding pressure-increment equation has zero mean. By [F3], there is exactly one mean-zero $ϕ^{n + 1}$ solving that equation.

4.1F1F2F3step 3.1algebra∎

The velocity and pressure updates are explicit after the unique predictor and pressure increment. Since $D_{h} G_{h} = Δ_{h}$ , the corrected divergence is $D_{h} \tilde{u} - k Δ_{h} ϕ = 0$ in backward Euler and $D_{h} \tilde{u} - (2 k / 3) Δ_{h} ϕ = 0$ in BDF2. The BDF2 rotational pressure correction subtracts $ν D_{h} \tilde{u}$ , which has zero mean by compatibility; thus both pressure updates stay in $Q_{h}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility

Statement

On the grid of Periodic staggered-grid velocity and pressure spaces, let $Δ_{h}^{- 1} : Q_{h} \to Q_{h}$ be the inverse from Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound. Define $P_{h} : = I - G_{h} Δ_{h}^{- 1} D_{h} : V_{h} \to V_{h} .$ Then $P_{h}$ is the mass-orthogonal projector onto $Z_{h} = \ker D_{h}$ . In the backward-Euler algorithm Backward-Euler incremental pressure correction on a periodic staggered grid, the velocity correction satisfies $u^{n + 1} = P_{h} {\tilde{u}}^{n + 1}$ and $D_{h} u^{n + 1} = 0$ exactly.

Facts & Assumptions

Given: The periodic staggered-grid spaces and operators, their mass pairings, a mean-zero pressure solve, and the backward-Euler correction.

[F1]

$D_{h} v$ is mean-zero, and $Δ_{h}$ is invertible on $Q_{h}$ (Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound).

[F2]

The backward-Euler algorithm sets $Δ_{h} ϕ^{n + 1} = k^{- 1} D_{h} {\tilde{u}}^{n + 1}$ and $u^{n + 1} = {\tilde{u}}^{n + 1} - k G_{h} ϕ^{n + 1}$ (Backward-Euler incremental pressure correction on a periodic staggered grid).

[F3]

The forward gradient, backward divergence, and mass pairings are the explicit finite-array formulas in Periodic staggered-grid velocity and pressure spaces.

Proof

1.1F1algebra

For any $v \in V_{h}$ , let $q = Δ_{h}^{- 1} D_{h} v \in Q_{h}$ , which is defined by [F1]. Then $D_{h} P_{h} v = D_{h} v - D_{h} G_{h} q = D_{h} v - Δ_{h} q = 0.$ Thus $P_{h} v \in Z_{h}$ . If $v \in Z_{h}$ , then $D_{h} v = 0$ , so uniqueness of the mean-zero inverse gives $q = 0$ and $P_{h} v = v$ . Therefore $P_{h}$ is a projection with range exactly $Z_{h}$ .

2.1F1F3step 1.1algebra

For $v \in V_{h}$ and $z \in Z_{h}$ , put $q = Δ_{h}^{- 1} D_{h} v \in Q_{h}$ , so $v - P_{h} v = G_{h} q$ . Reindexing the periodic sums in [F3] gives $(z, v - P_{h} v)_{M_{h}} = h^{d - 1} \sum_{i, j} z_{i, j} (q_{j + e_{i}} - q_{j}) = - h^{d - 1} \sum_{j} q_{j} \sum_{i} (z_{i, j} - z_{i, j - e_{i}}) = - (q, D_{h} z)_{Q_{h}} = 0.$ Hence the residual $v - P_{h} v$ is mass-orthogonal to the divergence-free subspace. Together with step 1.1 this makes $P_{h}$ the mass-orthogonal projector onto $Z_{h}$ .

3.1F1F2step 1.1algebra∎

In the algorithm, [F2] and uniqueness of the pressure solve give $ϕ^{n + 1} = k^{- 1} Δ_{h}^{- 1} D_{h} {\tilde{u}}^{n + 1}$ . Substitution into the velocity correction yields $u^{n + 1} = {\tilde{u}}^{n + 1} - G_{h} Δ_{h}^{- 1} D_{h} {\tilde{u}}^{n + 1} = P_{h} {\tilde{u}}^{n + 1} .$ Step 1.1 then gives $D_{h} u^{n + 1} = 0$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Modified velocity-pressure energy identity for periodic pressure correction

Statement

For one step of Backward-Euler incremental pressure correction on a periodic staggered grid on Periodic staggered-grid velocity and pressure spaces, with $u^{n}$ divergence- free and $p^{n}, p^{n + 1} \in Q_{h}$ , the following identity holds exactly: $\begin{aligned} ∥ u^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} & + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2} + 2 ν k ∥ G_{h} {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} \\ + k^{2} (∥ G_{h} p^{n + 1} ∥_{M_{h}}^{2} - ∥ G_{h} p^{n} ∥_{M_{h}}^{2}) = 2 k (f^{n + 1}, {\tilde{u}}^{n + 1})_{M_{h}} . \end{aligned}$ This is a modified velocity-pressure energy identity for the stated periodic discretization; it makes no bounded-wall claim.

Facts & Assumptions

Given: The backward-Euler predictor, pressure increment and correction on the periodic staggered grid, with $ν > 0$ , $k > 0$ , $D_{h} u^{n} = 0$ , and $p^{n}, p^{n + 1} \in Q_{h}$ .

[F1]

The predictor equation, pressure equation, and pressure update are $({\tilde{u}}^{n + 1} - u^{n}) / k + ν A_{h} {\tilde{u}}^{n + 1} + G_{h} p^{n} = f^{n + 1}$ , $Δ_{h} ϕ^{n + 1} = k^{- 1} D_{h} {\tilde{u}}^{n + 1}$ , $u^{n + 1} = {\tilde{u}}^{n + 1} - k G_{h} ϕ^{n + 1}$ , and $p^{n + 1} = p^{n} + ϕ^{n + 1}$ (Backward-Euler incremental pressure correction on a periodic staggered grid).

[F2]

The discrete adjoint relation is $(G_{h} q, v)_{M_{h}} = - (q, D_{h} v)_{Q_{h}}$ , and the pressure Laplacian satisfies $(q, Δ_{h} r)_{Q_{h}} = - (G_{h} q, G_{h} r)_{M_{h}}$ (Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility).

[F3]

$P_{h}$ is a mass-orthogonal projector and $u^{n + 1} = P_{h} {\tilde{u}}^{n + 1}$ (Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility).

[F4]

$(A_{h} v, v)_{M_{h}} = ∥ G_{h} v ∥_{M_{h}}^{2}$ by the periodic componentwise Laplacian definition (Periodic staggered-grid velocity and pressure spaces).

Proof

1.1givenF1F4algebra

By finite periodic reindexing in each component and coordinate, $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} = ∥ G_{h} v ∥_{M_{h}}^{2}$ . Take the $M_{h}$ inner product of the predictor equation [F1] with $2 k {\tilde{u}}^{n + 1}$ . The identity $2 ({\tilde{u}}^{n + 1} - u^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = ∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2}$ and [F4] give $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2} + 2 ν k ∥ G_{h} {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} + 2 k (G_{h} p^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = 2 k (f^{n + 1}, {\tilde{u}}^{n + 1})_{M_{h}} .$

2.1F1F2algebrastep 1.1

By [F2] and the pressure equation in [F1], $2 k (G_{h} p^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = - 2 k (p^{n}, D_{h} {\tilde{u}}^{n + 1})_{Q_{h}} = - 2 k^{2} (p^{n}, Δ_{h} ϕ^{n + 1})_{Q_{h}} = 2 k^{2} (G_{h} p^{n}, G_{h} ϕ^{n + 1})_{M_{h}} .$ The last equality follows from the second identity in [F2].

3.1F1F3step 2.1algebra

Since $P_{h}$ is orthogonal and $u^{n + 1} = P_{h} {\tilde{u}}^{n + 1}$ by [F3], the projection residual ${\tilde{u}}^{n + 1} - u^{n + 1} = k G_{h} ϕ^{n + 1}$ is orthogonal to $u^{n + 1}$ . Pythagoras gives $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} = ∥ u^{n + 1} ∥_{M_{h}}^{2} + k^{2} ∥ G_{h} ϕ^{n + 1} ∥_{M_{h}}^{2} .$ The pressure update $p^{n + 1} = p^{n} + ϕ^{n + 1}$ then gives $k^{2} (∥ G_{h} p^{n + 1} ∥_{M_{h}}^{2} - ∥ G_{h} p^{n} ∥_{M_{h}}^{2}) = k^{2} ∥ G_{h} ϕ^{n + 1} ∥_{M_{h}}^{2} + 2 k^{2} (G_{h} p^{n}, G_{h} ϕ^{n + 1})_{M_{h}} .$

4.1step 1.1step 3.1algebra∎

Substitute the two identities from step 3.1 into step 1.1, replacing $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2}$ and the pressure cross term. The result is exactly the displayed modified energy identity.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

First-order velocity accuracy for smooth periodic Stokes data

Statement

Let $d \geq 1$ , $ν > 0$ , $T > 0$ , $N \geq 2$ , $h = 1 / N$ , and $k = T / M$ for an integer $M \geq 1$ . Suppose smooth periodic fields $u, p, f$ on $[0, T] \times T^{d}$ satisfy $u_{t} - ν Δ u + \nabla p = f, \nabla \cdot u = 0, \int_{T^{d}} p (t, x) d x = 0, \int_{T^{d}} f_{i} (t, x) d x = 0$ for every $i, t$ . Assume the time derivatives of $u$ through order two, the spatial derivatives of $u$ through order four, of $p$ through order three, and of $f$ through order two are continuous and bounded. Sample $u$ at the staggered faces and $p, f$ at their corresponding grid locations; subtract the discrete mean from each sampled pressure and force component. Start the backward-Euler incremental pressure-correction algorithm with

u_{h}^{0} = P_{h} R_{h} u (0), p_{h}^{0} = R_{h} p (0) - h^{d} \sum_{j \in J_{N}} (R_{h} p (0))_{j} .

Then its velocity satisfies $\max_{0 \leq n \leq M} ∥ u_{h}^{n} - R_{h} u (t_{n}) ∥_{M_{h}} \leq C_{T} (h^{2} + k), t_{n} = n k,$ where $C_{T}$ is independent of $h, k$ and depends only on $d, T, ν$ and the displayed smoothness bounds. The estimate is for velocity; no pressure-error rate is asserted.

Facts & Assumptions

Given: The smooth periodic Stokes solution and compatibility conditions above; the backward-Euler scheme with discrete mean-zero force and pressure; the staggered-grid mass inner products.

[F1]

The discrete gradient/divergence are negative adjoints, $P_{h}$ is the mass-orthogonal projector onto $\ker D_{h}$ , and the backward-Euler correction is $u_{h}^{n + 1} = P_{h} {\tilde{u}}_{h}^{n + 1}$ (Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility).

[F2]

The predictor matrix $I + k ν A_{h}$ is invertible, and the pressure increment solve is unique on mean-zero arrays (Well-definedness of the periodic pressure-correction solves).

[F3]

For sampled smooth fields, the discrete divergence, vector Laplacian, and pressure gradient have uniform $O (h^{2})$ mass-norm consistency, and zero-integral smooth data have $O (h^{2})$ discrete mean (Second-order consistency of staggered periodic sampling).

[F4]

The pressure Poincare estimate is $∥ q ∥_{Q_{h}} \leq \sqrt{d} ∥ G_{h} q ∥_{M_{h}}$ for mean-zero $q$ (Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound).

[F5]

$∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ in the real mass inner products (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F6]

The finite differences are the centred periodic operators and the mass inner products are the $h^{d}$ weighted sums (Periodic staggered-grid velocity and pressure spaces).

[F7]

For a differentiable finite-dimensional vector curve with integrable derivative, the integral of the derivative is its endpoint difference, and a bounded derivative gives a Lipschitz bound (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F8]

The Euclidean norm of a vector integral is at most the integral of its Euclidean norm (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

Proof

1.1givenF3F6

Let $R_{h} u (t)$ denote the face samples and $R_{h} p (t), R_{h} f (t)$ the cell and face samples, respectively. For a face array $z$ , write ${\overset{ˉ}{z}}_{i}$ for the discrete mean of component $i$ repeated on every $i$ -face. The input force is $f_{h} (t) = R_{h} f (t) - \overset{ˉ}{R_{h} f (t)}$ , so [F3] gives $∥ f_{h} (t) - R_{h} f (t) ∥_{M_{h}} \leq C h^{2}$ . Define the sampling residual $r_{h} (t) = R_{h} u_{t} (t) + ν A_{h} R_{h} u (t) + G_{h} R_{h} p (t) - R_{h} f (t) .$ The continuous Stokes equation and [F3] imply $∥ r_{h} (t) ∥_{M_{h}} \leq C h^{2}$ , uniformly in $t$ .

2.1F1F2F3F6step 1.1algebra

Define $v_{h} (t) = P_{h} R_{h} u (t)$ and let $L_{h} = - Δ_{h}$ on mean-zero cell arrays. All coordinate shifts commute, so $A_{h} G_{h} = G_{h} L_{h}$ and $D_{h} A_{h} = L_{h} D_{h}$ . Since $L_{h} = - Δ_{h}$ , it commutes with $Δ_{h}^{- 1}$ on $Q_{h}$ . Substitution in $P_{h} = I - G_{h} Δ_{h}^{- 1} D_{h}$ gives $P_{h} A_{h} = A_{h} P_{h}$ . Also $P_{h} G_{h} q = 0$ for every mean-zero $q$ , because $G_{h} q$ is orthogonal to $\ker D_{h}$ by the adjoint relation in [F1]. Applying $P_{h}$ to the residual equation in step 1.1 therefore yields $v_{h}^{'} (t) + ν A_{h} v_{h} (t) = P_{h} f_{h} (t) + ρ_{h} (t),$ where $ρ_{h} = P_{h} r_{h} + P_{h} (R_{h} f - f_{h})$ . By [F1], $∥ P_{h} ∥ \leq 1$ , and hence $∥ ρ_{h} (t) ∥_{M_{h}} \leq C h^{2}$ .

3.1F1F2step 2.1algebra

Applying $P_{h}$ to the predictor equation and using $P_{h} A_{h} = A_{h} P_{h}$ and $P_{h} G_{h} p_{h}^{n} = 0$ shows that the actual corrected velocity is exactly the backward-Euler step on $Z_{h} = \ker D_{h}$ : $(I + k ν A_{h}) u_{h}^{n + 1} = u_{h}^{n} + k P_{h} f_{h} (t_{n + 1}) .$ The projected reference $v_{h}$ obeys the differential equation in step 2.1.

4.1F1F2F7F8step 2.1step 3.1algebra

Put $e_{h}^{n} = u_{h}^{n} - v_{h} (t_{n})$ . Integrating $v_{h}^{'}$ over $[t_{n}, t_{n + 1}]$ and using the equation in step 2.1 gives $(I + k ν A_{h}) v_{h} (t_{n + 1}) = v_{h} (t_{n}) + k P_{h} f_{h} (t_{n + 1}) + k ρ_{h} (t_{n + 1}) + τ_{h}^{n + 1},$ where $τ_{h}^{n + 1} = v_{h} (t_{n + 1}) - v_{h} (t_{n}) - k v_{h}^{'} (t_{n + 1})$ . By [F7], $τ_{h}^{n + 1} = \int_{t_{n}}^{t_{n + 1}} (v_{h}^{'} (s) - v_{h}^{'} (t_{n + 1})) d s .$ The bounded-derivative clause of [F7] gives $∥ v_{h}^{'} (s) - v_{h}^{'} (t_{n + 1}) ∥_{M_{h}} \leq (t_{n + 1} - s) \sup_{t} ∥ v_{h}^{''} (t) ∥_{M_{h}}$ after the isometric identification $v \mapsto h^{d / 2} v$ with Euclidean coordinates. Applying [F8] yields $∥ τ_{h}^{n + 1} ∥_{M_{h}} \leq \int_{t_{n}}^{t_{n + 1}} (t_{n + 1} - s) \sup_{t} ∥ v_{h}^{''} (t) ∥_{M_{h}} d s = \frac{k^{2}}{2} \sup_{t} ∥ v_{h}^{''} (t) ∥_{M_{h}} \leq C k^{2},$ because $P_{h}$ is a contraction and $h^{d} N^{d} = 1$ bounds the weighted sample norm by the continuous supremum norm up to the fixed component factor.

5.1F1F2step 3.1step 4.1algebra

For any face vector $z$ , periodic summation by parts gives $(A_{h} z, z)_{M_{h}} \geq 0$ . Hence $∥ (I + k ν A_{h}) z ∥_{M_{h}}^{2} = ∥ z ∥_{M_{h}}^{2} + 2 k ν (A_{h} z, z)_{M_{h}} + k^{2} ν^{2} ∥ A_{h} z ∥_{M_{h}}^{2} \geq ∥ z ∥_{M_{h}}^{2} .$ Thus $(I + k ν A_{h})^{- 1}$ is a contraction. Subtracting the equations in steps 3.1–4.1 and applying this contraction gives $∥ e_{h}^{n + 1} ∥_{M_{h}} \leq ∥ e_{h}^{n} ∥_{M_{h}} + C k h^{2} + C k^{2} .$ Since $e_{h}^{0} = 0$ and $M = T / k$ , summing yields $\max_{n} ∥ e_{h}^{n} ∥_{M_{h}} \leq C_{T} (h^{2} + k)$ .

6.1F1F3F4F5step 5.1algebra∎

Finally, $u_{h}^{n} - R_{h} u (t_{n}) = e_{h}^{n} + (P_{h} R_{h} u (t_{n}) - R_{h} u (t_{n}))$ . The second term equals $- G_{h} ϕ$ with $ϕ = Δ_{h}^{- 1} D_{h} R_{h} u (t_{n})$ . By [F4] and [F5], $∥ G_{h} ϕ ∥_{M_{h}}^{2} = - (ϕ, Δ_{h} ϕ)_{Q_{h}} = - (ϕ, D_{h} R_{h} u)_{Q_{h}} \leq \sqrt{d} ∥ G_{h} ϕ ∥_{M_{h}} ∥ D_{h} R_{h} u ∥_{Q_{h}} .$ If $G_{h} ϕ = 0$ the bound is immediate; otherwise divide by its norm. Since $\nabla \cdot u = 0$ , [F3] gives $∥ D_{h} R_{h} u ∥_{Q_{h}} \leq C h^{2}$ , so the projection-sampling discrepancy is $O (h^{2})$ . Combining this with step 5.1 proves the stated estimate.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Steady MINI Stokes pressure-mass Schur bounds

Statement

Assume Countable Choice. Let $D \subset R^{d}$ , $d \in {2, 3}$ , be a bounded connected Lipschitz polygonal or polyhedral domain with a finite, nonempty conforming shape-regular affine simplicial mesh. Use the MINI spaces $(V_{h}, Q_{h})$ of The conforming P1-bubble/P1 mini element, the Stokes norms and bilinear forms of Stokes velocity and pressure spaces and weak forms, and the viscous velocity form $a_{ν} (v, w) = 2 ν \int_{D} ε (v) : ε (w), ν > 0.$ Let $A_{h}, B_{h}$ be the matrices of $a_{ν}$ and $b (v, q) = - \int_{D} q div v$ in any bases, and let $M_{Q}$ be the $L^{2} (D)$ pressure mass matrix. Use the symmetric saddle convention of Symmetric saddle-point system, Schur complement, and block preconditioners. With $S_{h} = B_{h} A_{h}^{- 1} B_{h}^{T}$ , the steady Stokes pressure Schur complement satisfies $\frac{(C_{F} C_{d i v} (D))^{- 2}}{ν (1 + d)} M_{Q} ⪯ S_{h} ⪯ \frac{d}{ν} M_{Q} .$ Here $C_{F}$ is the shape-regularity-uniform Fortin norm bound and $C_{d i v} (D)$ is the continuous divergence right-inverse bound in Uniform Fortin stability of the mini element. The result is for the steady velocity stiffness; it asserts no corresponding bound for an unsteady shifted Schur complement.

Facts & Assumptions

Given: Countable Choice, the MINI Stokes pair on the stated finite nonempty mesh family, the homogeneous Dirichlet strain identity, and the generic Schur estimate from inf-sup stability.

[F1]

The velocity norm is $∥ v ∥_{V} = ∥ \nabla v ∥_{L^{2} (D)}$ , the pressure norm is $∥ q ∥_{Q} = ∥ q ∥_{L^{2} (D)}$ , and the Stokes form is $b (v, q) = - \int_{D} q div v$ (Stokes velocity and pressure spaces and weak forms).

[F2]

The MINI pair consists of continuous piecewise affine zero-boundary velocities enriched by cell bubbles and continuous piecewise affine mean-zero pressures on a conforming shape-regular affine simplicial mesh (The conforming P1-bubble/P1 mini element).

[F3]

On a finite connected conforming mesh, $\dim V_{h} = d N_{v, 0} + d N_{K}$ and $\dim Q_{h} = N_{v} - 1$ (Local and global dimensions of the mini element).

[F4]

For $v \in H_{0}^{1} (D; R^{d})$ , $2 ∥ ε (v) ∥_{2}^{2} = ∥ \nabla v ∥_{2}^{2} + ∥ div v ∥_{2}^{2}$ (The homogeneous Dirichlet strain identity).

[F5]

The mini-element discrete inf-sup constant satisfies $β_{h} \geq (C_{F} C_{d i v} (D))^{- 1}$ , with $C_{F}$ independent of mesh size (Uniform Fortin stability of the mini element).

[F6]

If $α ∥ v ∥_{V}^{2} \leq a (v, v) \leq Λ ∥ v ∥_{V}^{2}$ , $∣ b (v, q) ∣ \leq C_{b} ∥ v ∥_{V} ∥ q ∥_{Q}$ , and the inf-sup constant is $β$ , then $(β^{2} / Λ) M_{Q} ⪯ B A^{- 1} B^{T} ⪯ (C_{b}^{2} / α) M_{Q}$ (Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling).

[F7]

The standard real coordinate pairing is an inner product and Cauchy--Schwarz holds for it (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F8]

Holder's integral inequality gives $∣ \int_{D} f g ∣ \leq ∥ f ∥_{L^{2} (D)} ∥ g ∥_{L^{2} (D)}$ (Holder's inequality for integrals, including the endpoint cases).

[F9]

The generic saddle system has positive finite velocity and pressure dimensions, an SPD velocity matrix, and a full-row-rank constraint matrix (Symmetric saddle-point system, Schur complement, and block preconditioners).

[F10]

The mini inf-sup constant is $β_{h} = \inf_{q_{h} \neq 0} \sup_{v_{h} \neq 0} b (v_{h}, q_{h}) / (∥ v_{h} ∥_{V} ∥ q_{h} ∥_{Q})$ (Fortin operators transfer the continuous inf-sup bound).

[A1]

Countable Choice is the only choice principle assumed in the MINI, Korn and Fortin supplier chain (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F1F3F4F6F7F9givenalgebra

The given finite nonempty mesh contains at least one $d$ -simplex, so $N_{K} \geq 1$ and $N_{v} \geq d + 1$ . The dimension formulas [F3] give $\dim V_{h} \geq d > 0$ and $\dim Q_{h} \geq d > 0$ , verifying the dimension hypotheses in [F9]. For every nonzero pressure coefficient vector $q$ , $q^{T} M_{Q} q = ∥ q_{h} ∥_{2}^{2} > 0$ by the pressure norm in [F1], so $M_{Q}$ is SPD. For every $v \in V_{h}$ , [F4] and coordinate Cauchy--Schwarz [F7] give $∣ div v ∣^{2} \leq d ∣ \nabla v ∣^{2}$ pointwise. Integrating this inequality and using [F4] yields $ν ∥ \nabla v ∥_{2}^{2} \leq a_{ν} (v, v) \leq ν (1 + d) ∥ \nabla v ∥_{2}^{2} .$ The first inequality drops the nonnegative divergence term in [F4]; the second integrates the pointwise estimate. Thus $α = ν$ and $Λ = ν (1 + d)$ in [F6].

1.2F1F2F6F8algebra

For $v \in V_{h}$ and $q \in Q_{h}$ , Holder's inequality [F8] and the same pointwise divergence bound give $∣ b (v, q) ∣ \leq ∥ q ∥_{2} ∥ div v ∥_{2} \leq \sqrt{d} ∥ q ∥_{2} ∥ \nabla v ∥_{2} .$ Hence the continuity constant in [F6] is $C_{b} = \sqrt{d}$ .

2.1A1F5F6F9F10step 1.1step 1.2algebra∎

By [F5] and the definition [F10], the inf-sup constant in [F6] is at least $β = (C_{F} C_{d i v} (D))^{- 1} > 0$ . If $B_{h}^{T} q = 0$ for a nonzero pressure vector, then $b (v, q) = q^{T} B_{h} v = 0$ for every $v$ , making the supremum in [F10] zero and contradicting this positive lower bound. Thus $B_{h}$ has full row rank as required by [F9]. The lower bound in step 1.1 makes $A_{h}$ positive definite. Apply [F6] with the constants established in steps 1.1–1.2. Its lower coefficient is $(C_{F} C_{d i v} (D))^{- 2} / (ν (1 + d))$ and its upper coefficient is $d / ν$ . This is the asserted steady pressure-mass Schur bound.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

MINI Stokes block-preconditioned spectral intervals

Statement

Assume Countable Choice and the hypotheses of Steady MINI Stokes pressure-mass Schur bounds. Precondition the steady MINI saddle system symmetrically with $P = diag (A_{h}, ν^{- 1} M_{Q})$ , using the convention of Symmetric saddle-point system, Schur complement, and block preconditioners. Put $c_{-} = (C_{F} C_{d i v} (D))^{- 2} / (1 + d), c_{+} = d, λ_{\pm} (s) = \frac{1 \pm \sqrt{1 + 4 s}}{2} .$ Every eigenvalue of the Cholesky-symmetrically preconditioned matrix lies in $[λ_{-} (c_{+}), λ_{-} (c_{-})] \cup {1} \cup [λ_{+} (c_{-}), λ_{+} (c_{+})],$ where $1$ has multiplicity $\dim V_{h} - \dim Q_{h}$ (zero when the dimensions are equal). This interval is for the steady Stokes scaling $γ = 1 / ν$ ; it is not an iteration-count or solver-speed guarantee.

Facts & Assumptions

Given: Countable Choice, the steady MINI Schur bounds, and the spectrum formula for a symmetrically pressure-mass-preconditioned saddle system.

[F1]

If the eigenvalues of the normalized Schur matrix lie in $[a, b]$ , the preconditioned saddle eigenvalues are $λ_{-} (μ_{j})$ , $λ_{+} (μ_{j})$ , and $1$ with multiplicity $n - m$ (Spectrum of a symmetrically pressure-mass-preconditioned saddle system).

[F2]

The MINI steady Schur bounds are $(C_{F} C_{d i v} (D))^{- 2} (ν (1 + d))^{- 1} M_{Q} ⪯ S_{h} ⪯ d ν^{- 1} M_{Q}$ (Steady MINI Stokes pressure-mass Schur bounds).

[F3]

The finite MINI spaces satisfy $\dim V_{h} = d N_{v, 0} + d N_{K}$ and $\dim Q_{h} = N_{v} - 1$ (Local and global dimensions of the mini element).

[A1]

Countable Choice is the only choice principle used in the supplier chain (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F1F2F3algebra

The finite mesh contains a simplex, so [F3] gives $N_{K} \geq 1$ and $N_{v} \geq d + 1$ ; therefore both $V_{h}$ and $Q_{h}$ have positive dimensions as required by [F1]. Congruence of [F2] by $(ν^{- 1} M_{Q})^{- 1 / 2}$ shows that the normalized Schur eigenvalues $μ_{j}$ lie in $[(C_{F} C_{d i v} (D))^{- 2} / (1 + d), d] = [c_{-}, c_{+}] .$ Since $Q_{h}$ is nonzero and $M_{Q}$ is positive definite, [F2] also forces $c_{-} \leq c_{+}$ ; both endpoints are positive.

2.1F1step 1.1algebra∎

The derivative formulas $λ_{-}^{'} (s) = - (1 + 4 s)^{- 1 / 2} < 0$ and $λ_{+}^{'} (s) = (1 + 4 s)^{- 1 / 2} > 0$ for $s > 0$ show that the two eigenvalue branches map $[c_{-}, c_{+}]$ onto the two stated intervals. Applying [F1] therefore yields the displayed spectral enclosure and multiplicity.

5 · Examples, counterexamples and false statements

None yet.

Projection, pressure correction and saddle-point solvers

1 · Prerequisites

2 · Summary

Projection, pressure correction and saddle-point solvers

Periodic staggered-grid setting

Incremental correction and exact projection

BDF2 and rotational variants

Smooth periodic velocity accuracy

Saddle systems and conditional preconditioning bounds

Scope and source trail

3 · Logical flowchart

4 · Definitions, theorems and proofs

Periodic staggered-grid velocity and pressure spaces

Definition

Symmetric saddle-point system, Schur complement, and block preconditioners

Definition

Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound

Statement

Facts & Assumptions

Proof

Second-order consistency of staggered periodic sampling

Statement

Facts & Assumptions

Proof

Exact Schur solve for a full-row-rank symmetric saddle system

Statement

Facts & Assumptions

Proof

Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling

Statement

Facts & Assumptions

Proof

Backward-Euler incremental pressure correction on a periodic staggered grid

Definition

BDF2 standard and rotational incremental pressure-correction variants

Definition

Spectrum of a symmetrically pressure-mass-preconditioned saddle system

Statement

Facts & Assumptions

Proof

Well-definedness of the periodic pressure-correction solves

Statement

Facts & Assumptions

Proof

Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility

Statement

Facts & Assumptions

Proof

Modified velocity-pressure energy identity for periodic pressure correction

Statement

Facts & Assumptions

Proof

First-order velocity accuracy for smooth periodic Stokes data

Statement

Facts & Assumptions

Proof

Steady MINI Stokes pressure-mass Schur bounds

Statement

Facts & Assumptions

Proof

MINI Stokes block-preconditioned spectral intervals

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources