Homogenisation of a locally periodic medium with areas of low and high diffusivity

Abstract

We aim at understanding transport in porous materials consisting of regions with both high and low diffusivities. We apply a formal homogenisation procedure to the case wherethe heterogeneities are not arranged in a strictly periodic manner. The result is a two-scale model formulated inx-dependent Bochner spaces. We prove the weak solvability of the limit two-scale model for a prototypical advection–diffusion system of minimal size. A special feature of our analysis is that most of the basic estimates (positivity,L∞-bounds, uniqueness, energy inequality) are obtained in thex-dependent Bochner spaces.

Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

“…Then, we add all inequalities for ξ ∈ Z n , such that ε(ξ + Y ) ⊂ Ω, and obtain the first estimate in (17). The second estimate follows from the decomposition of Ω ε i into ∪ ξ∈Z n ε(ξ + Y i ) and Poincaré's inequality as in the previous estimate.…”

“…The striking thing is that in spite of the fact that the basic physical-chemistry of this relatively easy material is known [1], we have no control on how the microstructure changes (in time and space) and to which extent these spatio-temporal changes affect the observable macroscopic behavior of the material. The research reported here goes along the line open in [11], where a formal asymptotic expansion ansatz was used to derive macroscopic equations for a corrosion model, posed in a domain with locally-periodic microstructure (see [17] for a rigorous averaging approach of a reduced model defined in a domain with locallyperiodic microstructures). A two-scale convergence approach for periodic microstructures was studied in [10], while preliminary multiscale simulations are reported in [3].…”

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

“…By means of this estimate, we rigorously justified the formal homogenization asymptotics obtained in [37]. It is worth mentioning that the techniques developed for this context are applicable to a larger system of coupled partial differential equations posed in media with locally periodic heterogeneities.…”

“…In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity.…”

“…Our aim is twofold: On the one hand, we wish to justify rigorously the formal asymptotics expansions performed in [37], while on the other we wish to understand the error caused by replacing a heterogeneous solution by an approximation (averaged) estimate. To this end, we prove an upper bound for the convergence rate of the limit procedure → 0, where is a suitable scale parameter (see Section 2.1 for the definition of ).…”