High-contrast homogenization: Provider.PolynomialHomogenization.Root.Dirichlet.FoldAbsorption #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The two joins, and the eccentricity absorption into pEcc #
Two joins have to be supplied before the negative-Sobolev Dirichlet error clause's inhabitant can be assembled.
Join 1 — the two spellings of the fold factor. The energy collapse and
the output-factor bound are stated with (max 1 ecc) ^ p, while
eccentricityFoldFactor abar p is max 1 (ecc ^ p). They are
"equal exactly when 1 ≤ ecc, so the bridge lemma is one max_eq_right step".
In fact they are equal for every abar and every 0 ≤ p, with no
positive-definiteness hypothesis at all: below one, both sides are 1. So the
join costs nothing and carries no side condition.
Join 2 — absorbing an eccentricity power into pEcc. One step absorbs a
law-free residual constant Krest into the length factor Lg at the rate
κ. Its eccentricity companion — the move that lets pEcc be the sum of
the rate-leg and energy-leg exponents, which is the shape the restated hole
demands — is eccentricity_absorbed_into_fold below:
(C₀ · E ^ p) · (ε (x (Lg · fold p₀))) ^ κ ≤ C₀ · (ε (x (Lg · fold (p₀ + p/κ)))) ^ κ
The p/κ is exactly the κ-th root that the rate factor takes, and the sum is
performed by the root-interface's eccentricityFoldFactor_mul_le. Note
the shape of the enlarged length is x * (Lg * fold pEcc), not x * fold pEcc.
Join 1 #
Join 1. (max 1 ecc) ^ p and eccentricityFoldFactor abar p
(= max 1 (ecc ^ p)) are the same number for every nonnegative exponent — no
positive-definiteness hypothesis is needed.
Join 2 #
Join 2 — the eccentricity absorptions into pEcc. An eccentricity power
standing beside the constant is absorbed into the fold exponent of the enlarged
length, at the cost of p / κ added to pEcc. This is that step's
constant_absorbed_into_length for the fold slot rather than the Lg slot.