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

Abstract

In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fouriertype 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.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

“…As a concrete motivation of our proposed model (1), the authors in Conca et al (2004) delved into the chemical reactive flows through the exterior of a domain with distributed reactive obstacles, where the fractional (Langmuir kinetics) and polynomial (Freundlich kinetics) surface reactions were taken into account. Mathematically, such surface reactions posed on Γ ε read as −A(x/ε)∇u ε • n = ε β S (u ε ) for β ≥ 1, just like what has been studied in Khoa et al (2020). Note that even though we only consider in (1) the zero Neumann case of the internal boundary Γ ε , it is completely similar to adapt our analysis below to the surface reaction (using the same assumption as that of the volume one).…”

“…The presence of non-negative scalings stems from our mathematical concerns about the non-trivial asymptotic behaviors of u ε when ε tends to 0 and their corresponding rates. In fact, our most recent result in Khoa et al (2020) has shown that when α < 0 the macroscopic solution is identically zero after the homogenization process (ε → 0). The study of the variable scaled nonlinearities soon appeared in the works (Cabarrubias and Donato 2012;Conca et al 2004), where they considered the homogenization of elliptic problems with a scaled Robin boundary condition.…”

“…In this paper, we follow up on our earlier works (Khoa and Muntean 2016;Khoa 2017;Khoa et al 2020) that focus on the asymptotic analysis of semi-linear elliptic problems posed in perforated domains. This well-understood elliptic problem was applied in the studies of the heat transfer in composite materials and of the pressure and phase velocities in porous media flow.…”

“…Cf. Khoa et al (2020), the microscopic solution to the problem (P ε ) converges to a macroscopic function that solves a certain non-trivial homogenized problem since the scaling variable α ≥ 0 is eventually the main factor that determines the presence of the reaction term R at the macro-scale. In principle, the main purpose of homogenization is to unveil the macroscopic shape of u ε as it is less time-consuming than any approximation of u ε itself.…”

“…However, this result was only guaranteed when the diffusion must be very larger than the Lipschitz rate of reactions. Our next evolution in this area went to the work (Khoa et al 2020), where, for the first time, we addressed a linearization scheme for the weak solvability of a semi-linear microscopic system with real variable scalings. Since in those works, our interests were in the asymptotic analysis of the microscopic solution, we did not get through the computational standpoint of the homogenized system.…”