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

TL;DR

This paper investigates the homogenization of a microscopic semilinear elliptic equation in perforated domains, analyzing scaling effects, constructing correctors, and validating results through numerical examples.

Contribution

It introduces a linearization scheme for microscopic models, explores various scaling effects, and develops high-order corrector estimates for homogenization.

Findings

Established weak solvability of the microscopic model.

Derived macroscopic equations under compatible scalings.

Validated asymptotic analysis with numerical examples.

Abstract

In this paper, we consider a microscopic semilinear 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 a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space $H^1$, using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis.