CoarseGaugeResponse #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
Provider: the response functional under a constant skew shift of the field #
With the admissible class fixed by isAHarmonicGradient_sub_const_skew, the
shift a |-> a - hbar of ABK26 acts on the Chapter 2 response functional
Homogenization.Book.Ch02.responseJ by a shear of the load:
J(U, p, q; a - hbar) = J(U, p, q - hbar p; a) .
Indeed the response integrand
- (1/2) grad v . symm(a) grad v - p . a grad v + q . grad v
changes only in its middle term, by p . hbar grad v = - (hbar p) . grad v,
and its first term does not change at all because symm(a - hbar) = symm(a).
The two solution transfers transferSolutionSub and transferSolutionAdd
realize the class bijection at the level of
Homogenization.Book.Ch02.Solution, keeping the underlying H^1 function; the
flux L^2 bound they need is CoarseGraining's
Homogenization.Book.Ch02.Solution.flux_memVectorL2.
The class bijection #
An a-harmonic function, read as an (a - hbar)-harmonic function.
Equations
- Algsuperdiff.Section3.Provider.Diffusivity.ApproximateRecurrence.transferSolutionSub hskew hab u = { toH1 := u.toH1, isHarmonic := ⋯ }
Instances For
An (a - hbar)-harmonic function, read as an a-harmonic function.
Equations
- Algsuperdiff.Section3.Provider.Diffusivity.ApproximateRecurrence.transferSolutionAdd hskew hab v = { toH1 := v.toH1, isHarmonic := ⋯ }
Instances For
The sheared response #
The response integrand for a - hbar at load (p, q) is the response
integrand for a at the sheared load (p, q - hbar p).
The load shear identity for the Chapter 2 response functional.