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Problems including reactive transport processes through thin layers with a heterogeneous structure play an important role in many applications, especially from biosciences, medical sciences, geosciences, and material sciences. In our contribution, we consider a nonlinear reaction--diffusion equation in a domain consisting of two bulk-domains, which are separated by a thin layer with a periodic heterogeneous structure. The size of the heterogeneities and thickness of the layer are of order $\epsilon$, where the parameter $\epsilon$ is small compared to the length scale of the whole domain. In the limit $\epsilon \to 0$, when the thin layer reduces to an interface $\Sigma$ separating two bulk domains, a macroscopic model with effective interface conditions across $\Sigma$ is obtained. Here, the scaling of the microscopic model yields an effective reaction-diffusion equation at the interface $\Sigma$.
We investigate the quality of the approximation of the microscopic solution by means of the macroscopic one. In general, we cannot expect strong convergence of the gradients or high-order error estimates with respect to $\epsilon$. For such results we have to add additional corrector terms to the macroscopic solution which take into account the oscillations in the thin layer and also the coupling conditions between the bulk-regions and the layer. The construction of the approximations is made in two steps. Firstly, we add to the macroscopic solution in the thin layer a corrector of order $\epsilon$, which carries information about the oscillations in the layer. This leads to error estimates of order $\epsilon^{\frac{1}{2}}$ in the $H^1$-norms. To obtain a better estimate, in a second step, we add a corrector term of first order to the macroscopic solutions in the bulk-domains and an additional second order corrector to the macroscopic solution in the layer. In the layer, the correctors are obtained by products of the derivatives of the macroscopic solution and solutions of suitable cell problems on a bounded reference element, whereas in the bulk-domains the correctors include solution of boundary layer problems in infinite stripes.The resulting approximation leads to an error estimate of order $\epsilon$ in the $H^1$-norms. The techniques developed in this contribution for reaction-diffusion problems are a first step towards more challenging applications including e.g. advective transport or mechano-chemical interactions.
Andro Mikeli\'c gave significant contributions to the strategy of stepwise building up correctors to improve the effective approximations, especially of solutions to interface problems at the contact between a porous medium and a free fluid, see e.g., the fundamental paper:
W. J\"ager and \textbf{A. Mikeli\'c}: \textit{On the boundary conditions at the contact interface between a porous medium and a free fluid}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23:403-465 (1996).
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