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

  • Feppon, Florian
Publication Date
Mar 25, 2020
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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 unify the three possible kinds of limit problems derived by 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 to reconstruct the oscillating solution uε as a linear combination of the derivatives of its formal average u * ε weighted by suitable corrector tensors. The formal average is itself the 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 [52, 23], 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(ε 2K+4) in the L 2 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 coefficient-wise to either of the three classical homogenized regimes of the literature which arise when η is respectively lower, equivalent, or greater than the critical scaling η crit ∼ ε d/(d−2) .

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