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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.

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Sources

Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems (standard reference, not scraped)