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End date: 2019 × Start date: 2010 × Subject: degenerating diffusion × WGL Institute: WIAS ×

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    Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Reichelt, Sina; Thomas, Marita
    We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.
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    Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Reichelt, Sina
    We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms, and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1, other species may diffuse much slower, namely, with order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we prove that the error of its solution to the original solution is of order 1/2.