Manuscript Em03-D@1 · Em03-D-v1.tex · proposition · 450d878a

Proposition: Exact convergent transfer formula

A quantitative transfer estimate from advanced evolution › Reduced advanced operator and explicit constants · lines 1227–1274 · proof included or linked

Em03-D@1#prop:Neumann, Em03-D@1#eq:residualprobe

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The reduced advanced coupled solution is

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

Hence the reduced system test is

f_{ϵ} = - a (w_{ϵ})_{ψ} = f + r_{ϵ},

where

r_{ϵ} = - a [(I - ϵ^{2} A^{2})^{- 1} v_{A} - v_{A}] .

The reduced residual probe test is

(g_{ϵ})_{red} = g_{red} / ϵ + ϵ j_{ϵ}, j_{ϵ} = a A (I - ϵ^{2} A^{2})^{- 1} v_{A} .

Moreover,

∥ (w_{ϵ})_{ψ} - v_{A} ∥_{X} \leq \frac{ϵ ^{2} Λ ^{2}}{1 - z _{ϵ}} ∥ v_{A} ∥_{X} .

Proof. The two advanced equations give

(w_{ϵ})_{ϕ} = - ϵ A (w_{ϵ})_{ψ}, (w_{ϵ})_{ψ} = v_{A} - ϵ A (w_{ϵ})_{ϕ} .

Substitution gives the displayed inverse. Lemma lem:energy proves convergence in $X$ . The resulting solution has zero final data and satisfies the coupled equation. Cauchy uniqueness identifies it with the exact coupled advanced solution; smooth-source regularity makes it smooth. The test formulas and remainder estimate follow by substitution.

Displayed formulas in this passage

Em03-D@1#eq:residualprobe · line 1245
$(g_{ϵ})_{red} = g_{red} / ϵ + ϵ j_{ϵ}, j_{ϵ} = a A (I - ϵ^{2} A^{2})^{- 1} v_{A} .$
unlabeled · line 1230
$(w_{ϵ})_{ψ} = (I - ϵ^{2} A^{2})^{- 1} v_{A}, (w_{ϵ})_{ϕ} = - ϵ A (w_{ϵ})_{ψ} .$
unlabeled · line 1236
$f_{ϵ} = - a (w_{ϵ})_{ψ} = f + r_{ϵ},$
unlabeled · line 1240
$r_{ϵ} = - a [(I - ϵ^{2} A^{2})^{- 1} v_{A} - v_{A}] .$
unlabeled · line 1253
$∥ (w_{ϵ})_{ψ} - v_{A} ∥_{X} \leq \frac{ϵ ^{2} Λ ^{2}}{1 - z _{ϵ}} ∥ v_{A} ∥_{X} .$
unlabeled · line 1263
$(w_{ϵ})_{ϕ} = - ϵ A (w_{ϵ})_{ψ}, (w_{ϵ})_{ψ} = v_{A} - ϵ A (w_{ϵ})_{ϕ} .$

Proposition: Exact convergent transfer formula

Displayed formulas in this passage

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