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

  • *Corresponding author: Vo Anh Khoa

    *Corresponding author: Vo Anh Khoa 
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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  • 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.

    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]

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

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

    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)

    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 $

    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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