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10 High-order homogenization of the Poisson equation in a perforated periodic domain

From the book Optimization and Control for Partial Differential Equations

  • Florian Feppon


We derive high-order homogenized models for the Poisson problem in a cubic domain periodically perforated with holes, where Dirichlet boundary conditions are applied. These models have the potential to unify three possible kinds of limit problems derived in the literature for various asymptotic regimes (namely, the “unchanged” Poisson equation, the Poisson problem with a strange reaction term, and the zeroth-order limit problem) of the ratio η ≡ aϵ/ϵ between the size aϵ of the holes and the size ϵ of the periodic cell. The derivation relies on algebraic manipulations on formal two-scale power series in terms of ϵ and more particularly on the existence of a “criminal” ansatz, which allows us to reconstruct the oscillating solution uϵ as a linear combination of the derivatives of its formal average uϵ weighted by suitable corrector tensors. The formal average is itself a solution of a formal infinite-order homogenized equation. Classically, truncating the infinite-order homogenized equation yields in general an ill-posed model. Inspired by a variational method introduced in [23, 52], we derive, for any K ∈ ℕ, well-posed corrected homogenized equations of order 2K + 2, which yield approximations of the original solutions with an error of order O(ϵ2K+4) in the L2 norm. Finally, we find asymptotics of all homogenized tensors in the low volume fraction regime η → 0 and in dimension d ≥ 3. This allows us to show that our higher-order effective equations converge coefficientwise to either of the classical homogenized regimes of the literature, which arise when η is respectively equivalent to or greater than the critical scaling ηcrit ∼ ϵ2/(d−2).

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