Manuscript Em03-F@1 · Em03-F-v1.tex · section · 0d4d148b

Finite advanced estimates

A local scalar probe for the sine witness › Finite advanced estimates · lines 1803–1841

Em03-F@1#eq:localtransfer

For $ϵ^{2} Λ^{2} < 1$ , the exact reduced coupled solution is

(w_{ϵ})_{ψ} = (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A}, (w_{ϵ})_{ϕ} = - ϵ A_{p} (w_{ϵ})_{ψ} .

This follows from the two advanced equations; the convergent Neumann series solves them with zero final data, and uniqueness identifies it with the exact solution.

Consequently,

∥ k (B_{ϵ}) - l_{L} ∥_{U} \leq d_{p} (L, ϵ) := L_{p} \frac{ϵ ^{2} Λ ^{2}}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X},

where $k (B_{ϵ})$ denotes its reduced-test coefficient. The residual probe test has reduced form

(z_{ϵ})_{red} = z_{red} / ϵ + ϵ j_{ϵ},

j_{ϵ} = a_{p} A_{p} (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A},

∥ k (j_{ϵ}) ∥_{U} \leq L_{p} \frac{Λ}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X} .

Displayed formulas in this passage

Em03-F@1#eq:localtransfer · line 1817
$∥ k (B_{ϵ}) - l_{L} ∥_{U} \leq d_{p} (L, ϵ) := L_{p} \frac{ϵ ^{2} Λ ^{2}}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X},$
unlabeled · line 1806
$(w_{ϵ})_{ψ} = (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A}, (w_{ϵ})_{ϕ} = - ϵ A_{p} (w_{ϵ})_{ψ} .$
unlabeled · line 1827
$(z_{ϵ})_{red} = z_{red} / ϵ + ϵ j_{ϵ},$
unlabeled · line 1831
$j_{ϵ} = a_{p} A_{p} (I - ϵ^{2} A_{p}^{2})^{- 1} v_{A},$
unlabeled · line 1836
$∥ k (j_{ϵ}) ∥_{U} \leq L_{p} \frac{Λ}{1 - ϵ ^{2} Λ ^{2}} ∥ v_{A} ∥_{X} .$

Finite advanced estimates

Displayed formulas in this passage

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