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Subject: gradient folding operator ×

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    Corrector estimates for a thermo-diffusion model with weak thermal coupling
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Muntean, Adrian; Reichelt, Sina
    The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology weak thermal coupling refers here to the variable scaling in terms of the small homogenization parameter " of the heat conduction diffusion interaction terms, while the high-contrast is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with -independent estimates for the thermal and concentration fields and for their coupled fluxes.
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    Error estimates for elliptic equations with not exactly periodic coefficients
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Reichelt, Sina
    This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.