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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.

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Sources

J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows (standard reference, not scraped)
NPTEL, Lecture 5.4: Methods based on Coupling of Pressure and Velocity (standard reference, not scraped)