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A two-scale VMS closure attains its Schur defect bound

Example

Let each resolved and unresolved space be $R$ with the absolute-value norm (and give its dual the corresponding dual norm). Take the block matrix $(\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array})$ , load $(1, 0)$ , and closure inverse $T = 1 / 4$ . The exact and modeled coarse solutions differ by exactly the defect bound in Finite-dimensional VMS residual closure has an explicit Schur defect bound.

Facts & Assumptions

Given: The two-block VMS decomposition and approximate closure, with scalar block entries $A_{c c} = A_{f f} = 2$ , $A_{c f} = A_{f c} = 1$ , load $F_{c} = 1$ , $F_{f} = 0$ , and $T = 1 / 4$ .

[F1]

Finite-dimensional resolved–unresolved VMS split and residual closure: if $G = A_{f f}^{- 1}$ , the exact coarse Schur system is $(A_{c c} - A_{c f} G A_{f c}) u_{c} = F_{c} - A_{c f} G F_{f}$ and the modeled system replaces $G$ by $T$ .

[F2]

Finite-dimensional VMS residual closure has an explicit Schur defect bound: the coarse error is bounded by $∥ {\tilde{S}}^{- 1} ∥ ∥ A_{c f} ∥ ∥ T - G ∥ ∣ F_{f} - A_{f c} u_{c} ∣$ .

Verification

1.1F1givenalgebra

Here $G = 1 / 2$ , so the exact Schur coefficient is $2 - 1 / 2 = 3 / 2$ and the exact coarse solution is $u_{c} = 2 / 3$ . The exact unresolved component is $u_{f} = G (F_{f} - A_{f c} u_{c}) = - 1 / 3$ .

2.1F1step 1.1algebra

Replacing $G$ by $T = 1 / 4$ gives the approximate Schur coefficient $2 - 1 / 4 = 7 / 4$ , hence ${\tilde{u}}_{c} = 4 / 7$ , and ${\tilde{u}}_{f} = T (F_{f} - A_{f c} {\tilde{u}}_{c}) = - 1 / 7$ . The coarse error is $∣ 4 / 7 - 2 / 3 ∣ = 2 / 21$ .

3.1F1F2step 1.1step 2.1algebra∎

The bound in [F2] equals $(4 / 7) \cdot 1 \cdot (1 / 4) \cdot (2 / 3) = 2 / 21$ , because $∥ {\tilde{S}}^{- 1} ∥ = 4 / 7$ , $∣ A_{c f} ∣ = 1$ , $∣ T - G ∣ = 1 / 4$ , and $∣ F_{f} - A_{f c} u_{c} ∣ = 2 / 3$ . Thus the defect bound is attained exactly in this finite example.

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Sources

Thomas J. R. Hughes, Guglielmo Scovazzi, and Leopoldo P. Franca, Multiscale and Stabilized Methods, Encyclopedia of Computational Mechanics, ICES Report 04-18 (2004) (standard reference, not scraped)
Volker John, Numerical Methods for Convection-Dominated Problems, lecture notes, Chapter 5 (standard reference, not scraped)