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

Vo Anh Khoa; Thi Kim Thoa Thieu; Ekeoma Rowland Ijioma

doi:10.3934/dcdsb.2020190

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2021, Volume 26, Issue 5: 2451-2477. Doi: 10.3934/dcdsb.2020190

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

1.
Faculty of Sciences, Hasselt University, Campus Diepenbeek, BE3590 Diepenbeek, Belgium
2.
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, USA
3.
Department of Mathematics, Gran Sasso Science Institute, Viale Francesco Crispi 7, L'Aquila 67100, Italy
4.
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, Sweden
5.
Meiji Institute for Advanced Study of Mathematical Sciences, 4-21-1 Nakano, Nakano-ku, Tokyo, Japan

^*Corresponding author: Vo Anh Khoa
^*Corresponding author: Vo Anh Khoa

Received: July 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: May 2021

The work of V. A. K was partly supported by the Research Foundation-Flanders (FWO) under the project named "Approximations for forward and inverse reaction-diffusion problems related to cancer models". This work was also supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

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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.

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Mathematics Subject Classification: Primary: 35B27, 35C20, 35D30, 65M15.

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Figure 1. A schematic representation of a natural soil. The figure is followed from [31]

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Figure 2. A schematic representation of the scaling procedure within a natural soil and the corresponding sample periodically perforated domain with its unit cell

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Figure 3. Comparison between the homogenized solution and the microscopic solution for $ \varepsilon\in \left\{0.25, 0.05, 0.025\right\} $

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Figure 4. Behavior of the microscopic solution $ u_{\varepsilon} $ for the sub-cases $ \alpha = -1, \beta = 1 $ and $ \alpha = 1, \beta = -1 $ at $ \varepsilon = 0.25 $ (top) and $ \varepsilon = 0.025 $ (bottom)

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Figure 5. Convergence results in the $ \ell^{2} $-norm of $ u_{\varepsilon} $ in the microscopic domain for various combinations of the parameters $ \alpha, \beta $ and choices of $ \varepsilon $. First panel: $ \alpha = 1, \beta = 2 $. Second panel: $ \alpha = -1, \beta = 1 $ (dashed square) and $ \alpha = 1, \beta = -1 $ (solid diamond). Third panel: $ \alpha = 1, \beta = 1/2 $. Fourth panel: convergence at the micro-surfaces for $ \alpha = -2, C_{2} = 0 $

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Table 1. Numerical results in the $ \ell^{2} $-norm of $ u_{\varepsilon} $ at the micro-surfaces for $ \alpha = -2, C_{2} = 0 $

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Table 2. Numerical results in the $\ell^{2}$-norm of $u_{\varepsilon}$ at the micro-surfaces for $\alpha = -2, C_{2} = 0$

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