May  2021, 26(5): 2451-2477. doi: 10.3934/dcdsb.2020190

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

Received  August 2019 Revised  March 2020 Published  June 2020

Fund Project: 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

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.

Citation: Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

G. Allaire and M. Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var., 4 (1999), 209–243. doi: 10.1051/cocv:1999110.  Google Scholar

[3]

S. ArmstrongA. Gloria and T. Kuusi, Bounded correctors in almost periodic homogenization, Arch. Ration. Mech. Anal., 222 (2016), 393-426.  doi: 10.1007/s00205-016-1004-0.  Google Scholar

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G. A. Chechkin and T. A. Mel'nyk, Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses, Appl. Anal., 91 (2012), 1055-1095.  doi: 10.1080/00036811.2011.602634.  Google Scholar

[5]

G. A. Chechkin and A. L. Piatnitski, Homogenization of boundary-value problem in a locally periodic perforated domain, Appl. Anal., 71 (1999), 215-235.  doi: 10.1080/00036819908840714.  Google Scholar

[6]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences, 136, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar

[7]

C. DörlemannM. Heida and B. Schweizer, Transmission conditions for the Helmholtz-equation in perforated domains, Vietnam J. Math., 45 (2017), 241-253.  doi: 10.1007/s10013-016-0222-y.  Google Scholar

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F. FrankN. Ray and P. Knabner, Numerical investigation of homogenized Stokes-Nernst-Planck-Poisson systems, Comput. Vis. Sci., 14 (2011), 385-400.  doi: 10.1007/s00791-013-0189-0.  Google Scholar

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J. García-MeliánJ. D. Rossi and J. C. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Commun. Contemp. Math., 11 (2009), 585-613.  doi: 10.1142/S0219199709003508.  Google Scholar

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A. Gaudiello and T. Mel'nyk, Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary, J. Differential Equations, 265 (2018), 5419-5454.  doi: 10.1016/j.jde.2018.07.002.  Google Scholar

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

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G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286.   Google Scholar

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U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations, 92 (1991), 199-225.  doi: 10.1016/0022-0396(91)90047-D.  Google Scholar

[14]

J. Kačur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme, IMA J. Numer. Anal., 19 (1999), 119-145.  doi: 10.1093/imanum/19.1.119.  Google Scholar

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V. A. Khoa, A high-order corrector estimate for a semi-linear elliptic system in perforated domains, Comptes Rendus Mécanique, 345 (2017), 337-343.  doi: 10.1016/j.crme.2017.03.003.  Google Scholar

[16]

V. A. Khoa and A. Muntean, Asymptotic analysis of a semi-linear elliptic system in perforated domains: Well-posedness and correctors for the homogenization limit, J. Math. Anal. Appl., 439 (2016), 271-295.  doi: 10.1016/j.jmaa.2016.02.068.  Google Scholar

[17]

V. A. Khoa and A. Muntean, A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations, Appl. Math. Lett., 58 (2016), 103-109.  doi: 10.1016/j.aml.2016.02.009.  Google Scholar

[18]

V. A. Khoa and A. Muntean, Correctors justification for a Smoluchowski-Soret-Dufour model posed in perforated domains, preprint, arXiv: 1704.01790. Google Scholar

[19]

V. A. Khoa and A. Muntean, Corrector homogenization estimates for a non-stationary Stokes–Nernst–Planck–Poisson system in perforated domains, Commun. Math. Sci., 17 (2019), 705-738.  doi: 10.4310/CMS.2019.v17.n3.a6.  Google Scholar

[20]

S. Kim and K.-A. Lee, Higher order convergence rates in theory of homogenization Ⅲ: Viscous Hamilton-Jacobi equations, J. Differential Equations, 265 (2018), 5384-5418.  doi: 10.1016/j.jde.2018.07.003.  Google Scholar

[21]

O. KrehelT. Aiki and A. Muntean, Homogenization of a thermo-diffusion system with Smoluchowski interactions, Netw. Heterog. Media, 9 (2014), 739-762.  doi: 10.3934/nhm.2014.9.739.  Google Scholar

[22]

O. KrehelA. Muntean and P. Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregation and deposition, Advances in Water Resources, 86 (2015), 209-216.  doi: 10.1016/j.advwatres.2015.10.005.  Google Scholar

[23]

N. T. LongA. P. N. Dinh and T. N. Diem, Linear recursive schemes and asymptotic expansion associated with the Kirchoff–Carrier operator, J. Math. Anal. Appl., 267 (2002), 116-134.  doi: 10.1006/jmaa.2001.7755.  Google Scholar

[24]

T. A. Mel'nik, Asymptotic expansion of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube, J. Math. Sci., 75 (1995), 1646-1671.  doi: 10.1007/BF02368668.  Google Scholar

[25]

T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl., 351 (2009), 29-42.  doi: 10.1016/j.jmaa.2008.09.048.  Google Scholar

[26]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[27]

D. Onofrei and B. Vernescu, Error estimates in periodic homogenization with non-smooth coefficients, Asymptot. Anal., 54 (2007), 103-123.   Google Scholar

[28]

D. Onofrei and B. Vernescu, Asymptotic analysis of second-order boundary layer correctors, Appl. Anal., 91 (2012), 1097-1110.  doi: 10.1080/00036811.2011.616498.  Google Scholar

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar
[30]

G. Papanicolau, A. Bensoussan and J.-L. Lions, Asymptotic Analysis for Periodic Structures, North Holland, 1978.  Google Scholar

[31]

N. Ray, Colloidal Transport in Porous Media Modeling and Analysis, Ph.D thesis, University of Erlangen-Nuremberg, 2013. Google Scholar

[32]

N. RayA. Muntean and P. Knabner, Rigorous homogenization of a Stokes-Nernst-Planck-Poisson system, J. Math. Anal. Appl., 390 (2012), 374-393.  doi: 10.1016/j.jmaa.2012.01.052.  Google Scholar

[33]

N. RayT. van NoordenF. Frank and P. Knabner, Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure, Transp. Porous Media, 95 (2012), 669-696.  doi: 10.1007/s11242-012-0068-z.  Google Scholar

[34]

M. Schmuck, First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations, ZAMM Z. Angew. Math. Mech., 92 (2012), 304-319.  doi: 10.1002/zamm.201100003.  Google Scholar

[35]

M. Schmuck, New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials, J. Math. Phys., 54 (2013), 21pp. doi: 10.1063/1.4790656.  Google Scholar

[36]

M. Schmuck and S. Kalliadasis, Rate of convergence of general phase field equations in strongly heterogeneous media toward their homogenized limit, SIAM J. Appl. Math., 77 (2017), 1471–1492. doi: 10.1137/16M1079646.  Google Scholar

[37]

M. SchmuckM. PradasG. A. Pavliotis and S. Kalliadasis, Upscaled phase-field model for interfacial dynamics in strongly heterogeneous domains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 3705-3724.  doi: 10.1098/rspa.2012.0020.  Google Scholar

[38]

C. SchumacherF. Schwarzenberger and I. Veselić, A Glivenko–Cantelli theorem for almost additive functions on lattices, Stochastic Process. Appl., 127 (2017), 179-208.  doi: 10.1016/j.spa.2016.06.005.  Google Scholar

[39]

M. Slodi{č}ka, Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition, M2AN Math. Model. Numer. Anal., 35 (2001), 691-711.  doi: 10.1051/m2an:2001132.  Google Scholar

[40]

T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: ${L}_{2}$-operator error estimates, Mathematika, 59 (2013), 463-476.  doi: 10.1112/S0025579312001131.  Google Scholar

[41]

N. Triantafyllidis and S. Bardenhagen, The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models, J. Mech. Phys. Solids, 44 (1996), 1891-1928.  doi: 10.1016/0022-5096(96)00047-6.  Google Scholar

[42]

H. M. Versieux and M. Sarkis, Numerical boundary corrector for elliptic equations with rapidly oscillating periodic coefficients, Comm. Numer. Methods Engrg., 22 (2006), 577-589.  doi: 10.1002/cnm.834.  Google Scholar

[43]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511.  doi: 10.4213/rm9710.  Google Scholar

show all references

References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

G. Allaire and M. Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var., 4 (1999), 209–243. doi: 10.1051/cocv:1999110.  Google Scholar

[3]

S. ArmstrongA. Gloria and T. Kuusi, Bounded correctors in almost periodic homogenization, Arch. Ration. Mech. Anal., 222 (2016), 393-426.  doi: 10.1007/s00205-016-1004-0.  Google Scholar

[4]

G. A. Chechkin and T. A. Mel'nyk, Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses, Appl. Anal., 91 (2012), 1055-1095.  doi: 10.1080/00036811.2011.602634.  Google Scholar

[5]

G. A. Chechkin and A. L. Piatnitski, Homogenization of boundary-value problem in a locally periodic perforated domain, Appl. Anal., 71 (1999), 215-235.  doi: 10.1080/00036819908840714.  Google Scholar

[6]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences, 136, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar

[7]

C. DörlemannM. Heida and B. Schweizer, Transmission conditions for the Helmholtz-equation in perforated domains, Vietnam J. Math., 45 (2017), 241-253.  doi: 10.1007/s10013-016-0222-y.  Google Scholar

[8]

F. FrankN. Ray and P. Knabner, Numerical investigation of homogenized Stokes-Nernst-Planck-Poisson systems, Comput. Vis. Sci., 14 (2011), 385-400.  doi: 10.1007/s00791-013-0189-0.  Google Scholar

[9]

J. García-MeliánJ. D. Rossi and J. C. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Commun. Contemp. Math., 11 (2009), 585-613.  doi: 10.1142/S0219199709003508.  Google Scholar

[10]

A. Gaudiello and T. Mel'nyk, Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary, J. Differential Equations, 265 (2018), 5419-5454.  doi: 10.1016/j.jde.2018.07.002.  Google Scholar

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286.   Google Scholar

[13]

U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations, 92 (1991), 199-225.  doi: 10.1016/0022-0396(91)90047-D.  Google Scholar

[14]

J. Kačur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme, IMA J. Numer. Anal., 19 (1999), 119-145.  doi: 10.1093/imanum/19.1.119.  Google Scholar

[15]

V. A. Khoa, A high-order corrector estimate for a semi-linear elliptic system in perforated domains, Comptes Rendus Mécanique, 345 (2017), 337-343.  doi: 10.1016/j.crme.2017.03.003.  Google Scholar

[16]

V. A. Khoa and A. Muntean, Asymptotic analysis of a semi-linear elliptic system in perforated domains: Well-posedness and correctors for the homogenization limit, J. Math. Anal. Appl., 439 (2016), 271-295.  doi: 10.1016/j.jmaa.2016.02.068.  Google Scholar

[17]

V. A. Khoa and A. Muntean, A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations, Appl. Math. Lett., 58 (2016), 103-109.  doi: 10.1016/j.aml.2016.02.009.  Google Scholar

[18]

V. A. Khoa and A. Muntean, Correctors justification for a Smoluchowski-Soret-Dufour model posed in perforated domains, preprint, arXiv: 1704.01790. Google Scholar

[19]

V. A. Khoa and A. Muntean, Corrector homogenization estimates for a non-stationary Stokes–Nernst–Planck–Poisson system in perforated domains, Commun. Math. Sci., 17 (2019), 705-738.  doi: 10.4310/CMS.2019.v17.n3.a6.  Google Scholar

[20]

S. Kim and K.-A. Lee, Higher order convergence rates in theory of homogenization Ⅲ: Viscous Hamilton-Jacobi equations, J. Differential Equations, 265 (2018), 5384-5418.  doi: 10.1016/j.jde.2018.07.003.  Google Scholar

[21]

O. KrehelT. Aiki and A. Muntean, Homogenization of a thermo-diffusion system with Smoluchowski interactions, Netw. Heterog. Media, 9 (2014), 739-762.  doi: 10.3934/nhm.2014.9.739.  Google Scholar

[22]

O. KrehelA. Muntean and P. Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregation and deposition, Advances in Water Resources, 86 (2015), 209-216.  doi: 10.1016/j.advwatres.2015.10.005.  Google Scholar

[23]

N. T. LongA. P. N. Dinh and T. N. Diem, Linear recursive schemes and asymptotic expansion associated with the Kirchoff–Carrier operator, J. Math. Anal. Appl., 267 (2002), 116-134.  doi: 10.1006/jmaa.2001.7755.  Google Scholar

[24]

T. A. Mel'nik, Asymptotic expansion of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube, J. Math. Sci., 75 (1995), 1646-1671.  doi: 10.1007/BF02368668.  Google Scholar

[25]

T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl., 351 (2009), 29-42.  doi: 10.1016/j.jmaa.2008.09.048.  Google Scholar

[26]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[27]

D. Onofrei and B. Vernescu, Error estimates in periodic homogenization with non-smooth coefficients, Asymptot. Anal., 54 (2007), 103-123.   Google Scholar

[28]

D. Onofrei and B. Vernescu, Asymptotic analysis of second-order boundary layer correctors, Appl. Anal., 91 (2012), 1097-1110.  doi: 10.1080/00036811.2011.616498.  Google Scholar

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar
[30]

G. Papanicolau, A. Bensoussan and J.-L. Lions, Asymptotic Analysis for Periodic Structures, North Holland, 1978.  Google Scholar

[31]

N. Ray, Colloidal Transport in Porous Media Modeling and Analysis, Ph.D thesis, University of Erlangen-Nuremberg, 2013. Google Scholar

[32]

N. RayA. Muntean and P. Knabner, Rigorous homogenization of a Stokes-Nernst-Planck-Poisson system, J. Math. Anal. Appl., 390 (2012), 374-393.  doi: 10.1016/j.jmaa.2012.01.052.  Google Scholar

[33]

N. RayT. van NoordenF. Frank and P. Knabner, Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure, Transp. Porous Media, 95 (2012), 669-696.  doi: 10.1007/s11242-012-0068-z.  Google Scholar

[34]

M. Schmuck, First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations, ZAMM Z. Angew. Math. Mech., 92 (2012), 304-319.  doi: 10.1002/zamm.201100003.  Google Scholar

[35]

M. Schmuck, New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials, J. Math. Phys., 54 (2013), 21pp. doi: 10.1063/1.4790656.  Google Scholar

[36]

M. Schmuck and S. Kalliadasis, Rate of convergence of general phase field equations in strongly heterogeneous media toward their homogenized limit, SIAM J. Appl. Math., 77 (2017), 1471–1492. doi: 10.1137/16M1079646.  Google Scholar

[37]

M. SchmuckM. PradasG. A. Pavliotis and S. Kalliadasis, Upscaled phase-field model for interfacial dynamics in strongly heterogeneous domains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 3705-3724.  doi: 10.1098/rspa.2012.0020.  Google Scholar

[38]

C. SchumacherF. Schwarzenberger and I. Veselić, A Glivenko–Cantelli theorem for almost additive functions on lattices, Stochastic Process. Appl., 127 (2017), 179-208.  doi: 10.1016/j.spa.2016.06.005.  Google Scholar

[39]

M. Slodi{č}ka, Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition, M2AN Math. Model. Numer. Anal., 35 (2001), 691-711.  doi: 10.1051/m2an:2001132.  Google Scholar

[40]

T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: ${L}_{2}$-operator error estimates, Mathematika, 59 (2013), 463-476.  doi: 10.1112/S0025579312001131.  Google Scholar

[41]

N. Triantafyllidis and S. Bardenhagen, The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models, J. Mech. Phys. Solids, 44 (1996), 1891-1928.  doi: 10.1016/0022-5096(96)00047-6.  Google Scholar

[42]

H. M. Versieux and M. Sarkis, Numerical boundary corrector for elliptic equations with rapidly oscillating periodic coefficients, Comm. Numer. Methods Engrg., 22 (2006), 577-589.  doi: 10.1002/cnm.834.  Google Scholar

[43]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511.  doi: 10.4213/rm9710.  Google Scholar

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 $
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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