Based on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order , where describes the periodicity of the microstructure and depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.
E.Acerbi, V.Chiadò Piat, G.Dal Maso and D.Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal.18(5) (1992), 481–496. doi:10.1016/0362-546X(92)90015-7.
2.
A.Ainouz, Homogenization of a dual-permeability problem in two-component media with imperfect contact, Appl Math.60(2) (2015), 185–196. doi:10.1007/s10492-015-0090-x.
3.
E.Canon and J.N.Pernin, Homogénéisation d’un problème de diffusion en milieu composite avec barrière à l’interface, C R Acad Sci Paris Sèr I Math.325(1) (1997), 123–126. doi:10.1016/S0764-4442(97)83946-8.
4.
D.Cioranescu, A.Damlamian, P.Donato, G.Griso and R.Zaki, The periodic unfolding method in domains with holes, SIAM J Math Anal.44(2) (2012), 718–760. doi:10.1137/100817942.
5.
D.Cioranescu, A.Damlamian and G.Griso, The periodic unfolding method in homogenization, SIAM J Math Anal.40(4) (2008), 1585–1620. doi:10.1137/080713148.
6.
D.Cioranescu, P.Donato and R.Zaki, The periodic unfolding method in perforated domains, Port Math (NS)63(4) (2006), 467–496.
7.
D.Cioranescu and J.S.J.Paulin, Homogenization in open sets with holes, J Math Anal Appl.71(2) (1979), 590–607. doi:10.1016/0022-247X(79)90211-7.
8.
P.Donato, L.Faella and S.Monsurrò, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J Math Pures Appl (9)87(2) (2007), 119–143. doi:10.1016/j.matpur.2006.11.004.
9.
P.Donato and K.H.Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, NoDEA Nonlinear Differential Equations Appl.22(5) (2015), 1345–1380. doi:10.1007/s00030-015-0325-2.
10.
P.Donato, K.H.Le Nguyen and R.Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J Math Sci (NY)176(6) (2011), 891–927. Problems in mathematical analysis. No. 58. doi:10.1007/s10958-011-0443-2.
11.
P.Donato and S.Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance, Anal Appl (Singap)2(3) (2004), 247–273. doi:10.1142/S0219530504000345.
12.
P.Donato and I.Ţenţea, Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method, Bound Value Probl.2013(265) (2013), 14.
13.
W.Dreyer, C.Guhlke and R.Müller, Modeling of electrochemical double layers in thermodynamic non-equilibrium, Phys Chem Chem Phys.17 (2015), 27176–27194. doi:10.1039/C5CP03836G.
14.
T.Fatima, A.Muntean and M.Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl Anal.91(6) (2012), 1129–1154. doi:10.1080/00036811.2011.625016.
15.
G.Griso, Error estimate and unfolding for periodic homogenization, Asymptot Anal.40(3–4) (2004), 269–286.
16.
G.Griso, Interior error estimate for periodic homogenization, C R Math Acad Sci Paris.340(3) (2005), 251–254. doi:10.1016/j.crma.2004.10.027.
17.
G.Griso, Error estimates in periodic homogenization with a non-homogeneous Dirichlet condition, 2013, arXiv:13084110.
18.
P.Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.
19.
H.Hanke, Homogenization in gradient plasticity, Math Models Methods Appl Sci.21 (2011), 1651–1684. doi:10.1142/S0218202511005520.
20.
M.Heida, An extension of the stochastic two-scale convergence method and application, Asymptot Anal.72(1–2) (2011), 1–30.
21.
H.K.Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances, Appl Anal.75(3–4) (2000), 403–424. doi:10.1080/00036810008840857.
22.
E.C.Jose, Homogenization of a parabolic problem with an imperfect interface, Rev Roumaine Math Pures Appl.54(3) (2009), 189–222.
23.
M.Landstorfer, C.Guhlke and W.Dreyer, Theory and structure of the metal-electrolyte interface incorporating adsorption and solvation effects, Electrochimica Acta.201 (2016), 187–219. doi:10.1016/j.electacta.2016.03.013.
24.
K.H.Le Nguyen, Homogenization of heat transfer process in composite materials, J Elliptic Parabol Equ.1 (2015), 175–188. doi:10.1007/BF03377374.
25.
A.Mielke, S.Reichelt and M.Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw Heterog Media.9(2) (2014), 353–382. doi:10.3934/nhm.2014.9.353.
26.
S.Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv Math Sci Appl.13(1) (2003), 43–63.
27.
J.Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.
28.
S.Reichelt, Error estimates for elliptic equations with not-exactly periodic coefficients, Advances in Mathematical Sciences and Applications.25(1) (2016), 117–131.
29.
S.Reichelt, Two-scale homogenization of systems of nonlinear parabolic equations, Humboldt-Universität zu Berlin, 2015, http://edoc.hu-berlin.de/dissertationen/reichelt-sina-2015-11-27/PDF/reichelt.pdf.
30.
P.Simon and Y.Gogotsi, Materials for electrochemical capacitors, Nature Materials.7 (2008), 845–854. doi:10.1038/nmat2297.
