High-contrast homogenization: Entry.Response.Direct.TerminalDeficitCarrierBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Sign Facts and the Carrier Form of the Subcell Deficit Bound #
Continuing the terminal-optimizer replacement engine, this file records the two signs the
argument needs: restricting the terminal optimizer to a subcell gives an admissible competitor
there, so the subcell's own response dominates the restricted response, and the response of a
maximizer is itself nonnegative. It instantiates the pathwise deficit bound |½ ⨍_V ⟨∇u, symmPart(a) ∇u⟩ - J(V)| ≤ D + 2√(J(V) · D) on the aligned subcells of the adapted grid,
and transports it, with the nonnegativity of both sides, to the almost-everywhere elliptic
response coefficients respCoeff∓ F a. It also records the pathwise L^1(P) integrability of
the two coordinates of the weighted optimizer field that the cutoff-mean row's comparison
integrates. These are the sign and carrier facts consumed by the terminal-optimizer
replacement of p.response.transfer.
Nonnegativity of the subcell deficit and of the subcell response #
The terminal-optimizer replacement of p.response.transfer needs two signs. First, restricting the
terminal optimizer to a subcell gives an admissible competitor there, so its response value on the
subcell is at most the subcell's own response: the deficit is nonnegative. Second, the response of
a cell is nonnegative, because at a maximizer it is half the volume average of the nonnegative
variation-energy density of that maximizer.
The subcell deficit is nonnegative. Let V ⊆ U be open sets, a a coefficient elliptic
on V, and v a response maximizer on V for the load (p, r). Any a-harmonic function u
on U, restricted to the subcell V, is an admissible competitor there, so its response value on
V is at most the subcell response J(V; p, r; a). The restriction leaves the scalar response
integrand unchanged, so the deficit J(V) - ⨍_V g_V(u) is nonnegative. This is the sign input of
the terminal-optimizer replacement of p.response.transfer.
The subcell response is nonnegative. Let a be a coefficient elliptic on V and v a
response maximizer on V for the load (p, r). At a maximizer the response equals half the
volume average of the variation-energy density of v, and that density is nonnegative for an
elliptic coefficient, so J(V; p, r; a) ≥ 0. The side conditions are the integrability
hypotheses of the energy--response identity at the maximizer. This is the sign input of the
terminal-optimizer replacement of p.response.transfer.
The subcell deficit bound on the aligned cells of the adapted grid #
TerminalEnergyDeficitBound proves, for an abstract pair of open sets V ⊆ U, that the energy of
the
restricted optimizer of U on V is controlled by the response of V through the deficit of the
restricted optimizer there:
|½ ⨍_V ⟨∇u, symmPart(a) ∇u⟩ - J(V)| ≤ J(V) - ⨍_V g_U(u) + 2 √(J(V) · (J(V) - ⨍_V g_U(u))).
The terminal-optimizer replacement of p.response.transfer consumes this estimate on the concrete
pair consisting of an aligned subcell adaptedCellAtCenter q (t - n) w of the coarse scale
sitting inside
the terminal cell HighContrast.adaptedCell q t. This module records that instance: the subcell
is an
open bounded convex domain, it is contained in the terminal cell, and the ellipticity of b on the
terminal cell transfers to the subcell. The maximizer on the subcell and the finite-measure
instance are supplied by the caller, so the bound holds for whatever subcell maximizer is at hand.
The subcell deficit controls the terminal optimizer's energy on an aligned adapted
subcell. Let b be elliptic on the terminal cell HighContrast.adaptedCell q t, let u be
b-harmonic there, and let w ∈ triadicIndexBox d n, so that the depth-n subcell
adaptedCellAtCenter q (t - n) w is an open bounded convex domain contained in the terminal cell.
For
any response maximizer v on that subcell — with the subcell response and the deficit of the
restricted parent u — the terminal optimizer's energy on the subcell is controlled by
|½ ⨍ ⟨∇u, symmPart(b) ∇u⟩ - J| ≤ D + 2 √(J · D)
with J = J(adaptedCellAtCenter q (t - n) w; p, r; b) and
D = J - ⨍ g_{adaptedCell q t}(u). The caller supplies both the finite-measure instance and the
subcell maximizer together with the inequality relating them.
The subcell comparison at an almost-everywhere elliptic coefficient #
TerminalDeficitCarrierBound.abs_half_energy_adaptedCellAtCenter_sub_responseJ_le proves, for a
pointwise
elliptic coefficient b on the terminal cell and any b-harmonic u there, the subcell comparison
|½ ⨍_V g_b(u) − J(V)| ≤ D + 2√(J(V)·D), D = J(V) − ⨍_V g_U(u),
on each aligned subcell V = adaptedCellAtCenter q (t−n) w. The response coefficients
respCoeff∓ F a are elliptic only almost everywhere, so this module runs that estimate at a
pointwise elliptic representative and transports every quantity back across the a.e. replacement,
which leaves each of them invariant. It also records the two nonnegativities the row assembly
needs: 0 ≤ J(V) and 0 ≤ D.
The subcell comparison on an aligned adapted subcell, for the minus family, with no pointwise ellipticity hypothesis.
The subcell comparison on an aligned adapted subcell, for the plus family, with no pointwise ellipticity hypothesis.
Nonnegativity of the subcell response and of the subcell deficit, minus family.
Nonnegativity of the subcell response and of the subcell deficit, plus family.
Pathwise integrability of the weighted optimizer coordinates of the cutoff-mean split #
The cutoff-mean comparison of p.response.transfer integrates a coordinate of the doubled
optimizer state of the terminal optimizer against a bounded weight over a measurable subset of
the terminal cell. This module records the two pathwise integrability facts that step needs for
the recentred response coefficient respCoeffMinus F a and for its transposed twin
respCoeffPlus F a: the weighted first coordinate η · ∇v and the weighted second coordinate
η · (b ∇v), on any measurable V contained in the terminal cell.
The coefficient of the carrier is elliptic only almost everywhere, so the argument runs at the
pointwise elliptic representative supplied by
exists_elliptic_representative_respCell_respCoeffMinus / …Plus, extracts the coordinate
integrabilities of the representative through ResponseLinearIntegrabilityData, transports them
back across the almost-everywhere equality, restricts to V, and multiplies by the bounded
continuous weight.
Weighted optimizer coordinates for the minus response coefficient. For a terminal
maximizer uM a of the recentred coefficient respCoeffMinus F a on the terminal cell, and any
measurable V contained in that cell, both coordinates of the doubled optimizer state weighted by
a continuous η are integrable on V: the weighted gradient coordinate η · ∇v and the weighted
flux coordinate η · (b ∇v). The coefficient is transported from its almost-everywhere elliptic
representative, so the statement carries no pointwise ellipticity hypothesis.
Weighted optimizer coordinates for the plus response coefficient. The transposed twin of
integrableOn_weighted_optimizerField_respCell_respCoeffMinus: for a terminal maximizer uP a of
respCoeffPlus F a on the terminal cell and any measurable V contained in that cell, the
continuous-weight products η · ∇v and η · (b ∇v) are integrable on V.