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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 $D_{h} u^{n + 1} = 0$ .

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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)