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In this paper, we consider a microscopic semi-linear elliptic equation posed in periodically perforated domains and associated with the Fourier-type condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by choice of scale arguments and stabilization constant, to prove the weak solvability of the microscopic model. Asymptotic behaviors of solution with respect to the microscale parameter are thoroughly investigated in the second step, based on several cases of scalings. Following this path, we encapsulate such behaviors in the derivation of convergence rates in the microscopic domain. As an outcome of homogenization for multiscale elliptic problems, we derive the corresponding macroscopic equation whenever the scaling choices are compatible. Moreover, we prove a high-order corrector estimates for the homogenization limit in the energy space $H^1$; one of the main ingredients for the error analysis of the so-called multiscale finite element method. Our working techniques rely on a variational framework with distinctive types of two-scale asymptotic expansions to carry out energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis.

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit