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Homogenization, multiscale FEM and heterogeneous media — Examples

1 · Prerequisites

2 · Summary

These examples calculate the exact effective tensor for a layered scalar medium, show that its condition number can grow with contrast, evaluate a conditional FE-HMM error ledger with rational bounds, and calculate a nonzero one-dimensional resonance error at ratio $3 / 2$ . Each example states the precise coefficient, ratio, and hypotheses it uses. The ledger’s bounds are conditional certificates and do not assert a general FE-HMM rate.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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Layered medium: harmonic normal and arithmetic tangential means

Example

Let $d \geq 2$ , $0 < ρ < 1$ , and let $a (y) = a_{1}$ for $0 < y_{1} < ρ$ and $a (y) = a_{2}$ for $ρ < y_{1} < 1$ , extended periodically, where $a_{1}, a_{2} > 0$ . Then $A^{*} = diag ({(\frac{ρ}{a_{1}} + \frac{1 - ρ}{a_{2}})}^{- 1}, ρ a_{1} + (1 - ρ) a_{2}, \dots, ρ a_{1} + (1 - ρ) a_{2}) .$

Facts & Assumptions

Given: A scalar periodic coefficient layered only in the first coordinate, with positive phase values and phase fraction $ρ$ .

[F1]

Periodic correctors and effective tensor: columns of $A^{*}$ are averaged corrector fluxes.

[F2]

Existence and uniqueness of periodic cell correctors: the mean-zero cell correctors are unique.

[F3]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: iterated integration of the layered coefficient and smooth periodic test derivatives is valid.

[F4]

One-dimensional nonaligned micro sampling resonance bound: at phase zero and ratio one, the one-dimensional periodic mean-zero cell solution has averaged flux $(\int_{0}^{1} a (t)^{- 1} d t)^{- 1}$ .

Proof

1.1F2F1F3F4algebra

Set $c = (ρ / a_{1} + (1 - ρ) / a_{2})^{- 1}$ . Apply [F4] to the one-dimensional layered coefficient, with $λ = \min (a_{1}, a_{2})$ , $Λ = \max (a_{1}, a_{2})$ , phase zero and ratio one. Its mean-zero solution $w \in V_{#} (0, 1)$ satisfies $\int_{0}^{1} a (t) (1 + w^{'} (t)) v^{'} (t) d t = 0$ for every $v \in V_{#} (0, 1)$ and has mean flux $c$ . By the defining periodic closure, take smooth periodic $w_{ℓ} \to w$ in $H^{1} (0, 1)$ and subtract their means. Their lifts $W_{ℓ} (y) = w_{ℓ} (y_{1})$ are smooth periodic, and iterated integration gives $∥ W_{ℓ} - W_{k} ∥_{H^{1} (Y)} = ∥ w_{ℓ} - w_{k} ∥_{H^{1} (0, 1)}$ . Thus their $H^{1}$ limit is $χ_{1} (y) = w (y_{1}) \in V_{#} (Y)$ , with gradient $w^{'} (y_{1}) e_{1}$ . For every smooth periodic $v (y)$ , each slice $v (\cdot, y^{'})$ is an admissible one-dimensional cell test after subtracting its mean, so iterated integration gives $\int_{Y} a (y_{1}) (1 + w^{'} (y_{1})) \partial_{1} v (y) d y = 0$ . Density extends this identity to $V_{#} (Y)$ ; uniqueness [F2] identifies $χ_{1}$ as the first corrector. Its mean flux is $c e_{1}$ by [F4] and iterated integration.

2.1F3F1F2step 1.1algebra∎

For each $j \geq 2$ , $χ_{j} = 0$ solves the cell equation: for smooth periodic $v$ , iterated integration [F3] in $y_{j}$ gives $\int_{Y} a (y_{1}) \partial_{j} v = 0$ by equality of the opposite-face values, and density extends the identity to $V_{#}$ . Its mean flux is $\int_{Y} a (y_{1}) e_{j} = (ρ a_{1} + (1 - ρ) a_{2}) e_{j}$ . These columns yield the displayed diagonal matrix.

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High contrast destroys contrast-independent conditioning

Example

In dimension $d \geq 2$ , let $a_{κ} (y_{1}) = κ$ on the first half-period and $1$ on the second, with $0 < κ \leq 1$ . Then $A_{κ}^{*} = diag (\frac{2 κ}{1 + κ}, \frac{1 + κ}{2}, \dots, \frac{1 + κ}{2}), {cond}_{2} (A_{κ}^{*}) = \frac{(1 + κ)^{2}}{4 κ} ⟶ \infty (κ ↓ 0) .$ Thus pointwise ellipticity of each fixed coefficient does not give a contrast-independent condition number.

Facts & Assumptions

Given: A layered periodic scalar coefficient with positive values $κ$ and $1$ on equal fractions, where each fixed $κ$ is uniformly elliptic with $λ = κ$ and $Λ = 1$ .

[F1]

Symmetry and uniform ellipticity of the effective tensor: $A^{*}$ is symmetric positive definite for every fixed $κ > 0$ .

[F2]

Periodic correctors and effective tensor: the effective matrix is the mean corrector flux.

[F3]

One-dimensional nonaligned micro sampling resonance bound: the mean-zero harmonic-mean corrector for the one-dimensional profile belongs to $V_{#}$ ; its smooth periodic approximants lift to the higher-dimensional cell.

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: iterated integration over the unit cell is valid for the layered coefficient times a smooth periodic test derivative.

Proof

1.1F1F2F3F4algebra

The normal corrector has derivative $χ_{1}^{'} = c / a_{κ} - 1$ with $c = (\int_{0}^{1} a_{κ}^{- 1})^{- 1} = 2 κ / (1 + κ)$ ; this derivative has zero mean, and its periodic primitive gives constant normal flux $c$ . With $λ = κ$ and $Λ = 1$ , [F3] supplies the one-dimensional mean-zero corrector; lifting its smooth periodic approximants in the first coordinate gives the required member of $V_{#} (Y)$ . For each tangential coordinate $j \geq 2$ , zero is the corrector: for every smooth periodic $v$ , Tonelli [F4] and the one-dimensional FTC in $y_{j}$ give $\int_{Y} a_{κ} (y_{1}) \partial_{j} v (y) d y = 0$ , and density extends this to $V_{#}$ . Its mean flux is $\int_{0}^{1} a_{κ} = (1 + κ) / 2$ . These correctors yield the stated diagonal entries.

2.1F1step 1.1algebra∎

Since $0 < κ \leq 1$ , the tangential eigenvalue is at least the normal one: $(1 + κ)^{2} - 4 κ = (1 - κ)^{2} \geq 0$ . The spectral condition number is therefore their ratio, $(1 + κ)^{2} / (4 κ)$ , which diverges as $κ ↓ 0$ . Every fixed $κ$ remains positive and satisfies [F1]; only a uniform-in-contrast condition number fails.

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A nonaligned one-dimensional micro window has resonance error

Example

Let $a (y) = 1$ on $[0, 1 / 2)$ and $a (y) = 2$ on $[1 / 2, 1)$ , extended with period one. At phase $θ = 0$ and nonaligned ratio $r = δ / ϵ = 3 / 2$ , the exact sampled harmonic tensor is $A_{r} = 6 / 5$ , whereas $A^{*} = 4 / 3$ . The discrepancy is $2 / 15 > 0$ even though the continuum coefficient is periodic.

Facts & Assumptions

Given: The two-phase periodic scalar coefficient, phase zero, and sample ratio $r = 3 / 2$ .

[F1]

One-dimensional nonaligned micro sampling resonance bound: the sampled tensor is the reciprocal of the average reciprocal coefficient and obeys $∣ A_{r} - A^{*} ∣ \leq 2 Λ^{2} / (λ r)$ .

[F2]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: the affine pullback $y \mapsto 3 y / 2$ preserves measurability and null sets, so the sampled reciprocal integral is well defined on the stated coefficient class.

Proof

1.1F1F2algebra

By [F2], the sample pullback and its reciprocal are measurable on $[0, 1]$ . Over $0 \leq t \leq 3 / 2$ , the reciprocal coefficient equals $1$ for lengths $1 / 2$ , $1 / 2$ , and $1 / 2$ on the successive intervals $[0, 1 / 2)$ , $[1 / 2, 1)$ , and $[1, 3 / 2)$ , except that its value on the middle interval is $1 / 2$ . Thus its integral is $1 / 2 + 1 / 4 + 1 / 2 = 5 / 4$ ; dividing by $r = 3 / 2$ gives $\int_{0}^{1} a (3 y / 2)^{- 1} d y = 5 / 6$ , so $A_{r} = 6 / 5$ . The full-period reciprocal mean is $(1 + 1 / 2) / 2 = 3 / 4$ , hence $A^{*} = 4 / 3$ .

2.1F1step 1.1algebra∎

Therefore $∣ A_{r} - A^{*} ∣ = ∣ 6 / 5 - 4 / 3 ∣ = 2 / 15$ . Here $λ = 1$ , $Λ = 2$ , and $1 / r = 2 / 3$ , so the bound in [F1] is $2 \cdot 4 \cdot (2 / 3) = 16 / 3$ ; indeed $2 / 15 \leq 16 / 3$ . The example verifies both nonzero resonance and the stated conservative estimate.

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FE-HMM energy error ledger with distinct components

Example

Suppose a verified FE-HMM ledger has ellipticity bounds $λ = 1$ , $Λ = 2$ , macro best-approximation bound $η_{H} \leq 1 / 10$ , micro tensor discrepancy bound $Δ_{m i c r o} \leq 1 / 20$ , sampling discrepancy bound $Δ_{r e s} \leq 1 / 20$ , and homogenized energy norm $∥ \nabla u_{0} ∥_{2} \leq 1$ . Then the macro theorem certifies $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{4}{9} .$ These are conditional certified inputs for the ledger; the example makes no claim that they arise for every coefficient, scale ratio, or mesh.

Facts & Assumptions

Given: The six rational bounds stated above, with the micro and sampling discrepancies measured against the exact sampled tensor and $A^{*}$ respectively.

[F1]

FE-HMM macro, micro, and sampling error separation: if $Δ = Δ_{m i c r o} + Δ_{r e s} < λ$ , then $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{λ + Λ}{λ - Δ} η_{H} + \frac{Δ}{λ - Δ} ∥ \nabla u_{0} ∥_{2} .$

[F2]

Energy identity for FE-HMM micro tensor error: the micro discrepancy can be bounded by the squared best periodic P1 gradient approximation error.

Proof

1.1F1algebra

Here $Δ \leq 1 / 20 + 1 / 20 = 1 / 10 < 1 = λ$ , so the theorem applies. Its two contributions are bounded by $\frac{1 + 2}{1 - 1 / 10} \cdot \frac{1}{10} = \frac{1}{3}, \frac{1 / 10}{1 - 1 / 10} \cdot 1 = \frac{1}{9} .$ Their sum is $1 / 3 + 1 / 9 = 4 / 9$ . The first contribution uses the macro best approximation; the second is the sum of the distinct micro and resonance discrepancy certificates.

2.1F2F1step 1.1algebra∎

The micro part may be justified by an upper bound $\max_{K} η_{K}^{2} \leq 1 / 20$ from [F2], while the resonance part is separately bounded by $1 / 20$ ; if every ratio is integer, the resonance part is exactly zero by the aligned-sampling lemma cited in [F1]. This illustrates the ledger’s distinct inputs without asserting an unconditional rate.