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FE-HMM macro, micro, and sampling error separation

Statement

Let $U_{H}$ be the FE-HMM solution from Periodic FE-HMM with simplicial macro and periodic P1 micro meshes on a matching macro triangulation of $Q = (0, 1)^{2}$ , and let $u_{0} \in H_{0}^{1} (Q)$ solve the homogenized equation with coefficient $A^{*}$ . For each macro cell let $A_{K}^{e x}$ be the exact periodic-coupling sampled tensor and $A_{K}^{h}$ its discrete micro tensor. Define $Δ_{m i c r o} = \max_{K} ∥ A_{K}^{h} - A_{K}^{e x} ∥_{o p}, Δ_{r e s} = \max_{K} ∥ A_{K}^{e x} - A^{*} ∥_{o p}, Δ = Δ_{m i c r o} + Δ_{r e s},$ and $η_{H} = \inf_{V \in V_{H}} ∥ \nabla (u_{0} - V) ∥_{L^{2} (Q)}$ . If $Δ < λ$ , then $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{λ + Λ}{λ - Δ} η_{H} + \frac{Δ}{λ - Δ} ∥ \nabla u_{0} ∥_{2} .$ The first term is macro best approximation; the second is explicitly split by $Δ_{m i c r o}$ and $Δ_{r e s}$ . By Energy identity for FE-HMM micro tensor error, $Δ_{m i c r o} \leq \max_{K} η_{K}^{2}$ . If every micro ratio is an integer, No sampling resonance for period-aligned periodic micro cells gives $Δ_{r e s} = 0$ . The condition $Δ < λ$ is a sufficient perturbation condition; no arbitrary-scale accuracy is asserted.

Facts & Assumptions

Given: A finite conforming macro mesh of the unit square, its continuous zero-boundary P1 space, positive cell weights summing to area, uniformly elliptic symmetric local micro tensors, a homogenized solution, and the exact sampled tensors.

[F1]

Periodic FE-HMM with simplicial macro and periodic P1 micro meshes: $B_{H} (U, V) = \sum_{K} ∣ K ∣ \nabla U ∣_{K} \cdot A_{K}^{h} \nabla V ∣_{K}$ and the macro equation has right side $\int_{Q} f V$ .

[F2]

Well-posedness of the finite FE-HMM micro solves: each $A_{K}^{h}$ is symmetric and $λ$ -elliptic; the macro system has a unique solution.

[F3]

Energy identity for FE-HMM micro tensor error: $∥ A_{K}^{h} - A_{K}^{e x} ∥_{o p} \leq η_{K}^{2}$ .

[F4]

No sampling resonance for period-aligned periodic micro cells: integer period ratios give $A_{K}^{e x} = A^{*}$ .

[F5]

Symmetry and uniform ellipticity of the effective tensor: $A^{*}$ is symmetric with $λ I \leq A^{*} \leq Λ I$ .

[F6]

Compatible 2D newest-vertex-bisection meshes and Matching affine simplicial meshes and shape regularity: the selected macro triangulation is finite, conforming, and covers $\overline{Q}$ with disjoint cell interiors.

[F7]

Continuous matching piecewise polynomials belong to W1p and Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare: $V_{H} \subset H_{0}^{1} (Q)$ ; Separable H1-zero space and compact L2 inclusion on a cube gives a norm controlled by the gradient.

Proof

1.1F2F5F1F6F7algebra

By [F2] the macro solution and all local tensors are well defined. Write $D_{K} = A_{K}^{h} - A^{*}$ and $D_{0} = \max_{K} ∥ D_{K} ∥_{o p}$ . The triangle inequality gives $D_{0} \leq Δ$ . For macro functions $V, W$ , piecewise constancy of their P1 gradients and Cauchy--Schwarz over the finite cells imply $∣ B_{H} (V, W) - \int_{Q} A^{*} \nabla V \cdot \nabla W ∣ \leq D_{0} ∥ \nabla V ∥_{2} ∥ \nabla W ∥_{2} .$ By [F5] the reference form is at least $λ ∥ \nabla V ∥_{2}^{2}$ , so [F1] and $D_{0} \leq Δ < λ$ also give $B_{H} (V, V) \geq (λ - Δ) ∥ \nabla V ∥_{2}^{2}$ .

2.1F5step 1.1F1algebra

Fix any $V_{H} \in V_{H}$ and put $e_{H} = U_{H} - V_{H}$ . Since both macro and homogenized equations have the same load, for every test in $V_{H}$ , $B_{H} (e_{H}, e_{H}) = \int_{Q} A^{*} \nabla (u_{0} - V_{H}) \cdot \nabla e_{H} + (\int_{Q} A^{*} \nabla V_{H} \cdot \nabla e_{H} - B_{H} (V_{H}, e_{H})) .$ The right side has magnitude at most $(Λ ∥ \nabla (u_{0} - V_{H}) ∥_{2} + D_{0} ∥ \nabla V_{H} ∥_{2}) ∥ \nabla e_{H} ∥_{2}$ by [F5] and step 1.1. Combining with coercivity in step 1.1 and $∥ \nabla V_{H} ∥_{2} \leq ∥ \nabla u_{0} ∥_{2} + ∥ \nabla (u_{0} - V_{H}) ∥_{2}$ gives $∥ \nabla (U_{H} - u_{0}) ∥_{2} \leq \frac{λ + Λ}{λ - D_{0}} ∥ \nabla (u_{0} - V_{H}) ∥_{2} + \frac{D_{0}}{λ - D_{0}} ∥ \nabla u_{0} ∥_{2} .$ This also holds if $e_{H} = 0$ , when division is unnecessary.

3.1F3F4step 2.1algebra∎

Take a sequence $V_{H}$ whose approximation errors decrease to $η_{H}$ ; finite-dimensionality ensures the infimum is finite, and the displayed inequality passes to the limit. The two coefficient functions are increasing for $0 \leq D_{0} \leq Δ < λ$ , so replace $D_{0}$ by $Δ$ to obtain the stated estimate. Finally, the exact decomposition $A_{K}^{h} - A^{*} = (A_{K}^{h} - A_{K}^{e x}) + (A_{K}^{e x} - A^{*})$ gives the separate micro and resonance terms, with their micro-space and aligned-sampling interpretations from [F3]–[F4].

Depends on

Used by

FE-HMM energy error ledger with distinct components Example

Dependency tree · two levels

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Sources

Patrick Ciarlet Jr. and Christian Stohrer, Finite-element heterogeneous multiscale method for the Helmholtz equation (standard reference, not scraped)