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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.

Depends on

Used by

MINI Stokes block-preconditioned spectral intervals Corollary

Dependency tree · two levels

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Sources

Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems (standard reference, not scraped)
J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows (standard reference, not scraped)