Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T04:59:33.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

Published online by Cambridge University Press:  02 April 2013

A. MUNTEAN  and
T. L. VAN NOORDEN
A. MUNTEAN
Affiliation:
CASA – Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: [email protected] Institute for Complex Molecular Systems (ICMS), Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
T. L. VAN NOORDEN
Affiliation:
Chair of Applied Mathematics 1, Department of Mathematics, University of Erlangen-Nürnberg, Martensstraße 3, Erlangen 91058, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove an upper bound for the convergence rate of the homogenization limit ε → 0 for a linear transmission problem for a advection–diffusion(–reaction) system posed in areas with low and high diffusivity, where ε is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493–516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.

Keywords

Corrector estimatesTransmission conditionHomogenizationMicro–macro transportReaction–diffusion system in heterogeneous materials
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 5 , October 2013 , pp. 657 - 677
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avellandeda, M. (1996) Homogenization and renormalization: The mathematics of multi-scale random media and turbulent diffusion. Dynamical Systems and Probabilistic Methods in Partial Differential Equations, Lectures in Applied Mathematics, Vol. 31. AMS, Providence, RI, pp. 241–268.Google Scholar
[2]Baffico, L. & Conca, C. (1999) Homogenization of a transmission problem in solid mechanics. J. Math. Anal. Appl. 233, 659–680.Google Scholar
[3]Bal, G. & Jiang, W. (2010) Homogenization and corrector theory for linear transport in random media. Discr. Cont. Dyn. Syst. 28, 1311–1343.Google Scholar
[4]Belyaev, A. Y. & Efendiev, Y. R. (1996) Homogenization of the Stokes equations with a random potential. Math. Notes 59, 361–372.CrossRefGoogle Scholar
[5]Bensoussan, A., Lions, J. L. & Papanicolaou, G. 1978 Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, Vol. 5. North-Holland, Amsterdam, Netherlands.Google Scholar
[6]Bourgeat, A., Mikelic, A. & Piatnitski, A. (2003) On the double porosity model of a single phase flow in random media. Asymptotic Anal. 34, 311–332.Google Scholar
[7]Capriz, G. & Mazzini, G. (1998) Invariance and balance in continuum mechanics. Nonlinear Analysis and Continuum Mechanics: Papers for the 65th Birthday of James Serrin, Springer Verlag, Berlin, Germany, pp. 27–36.Google Scholar
[8]Chechkin, G. A., D'Apice, C. & Maio, U. D. (2010) On the rate of convergence of solutions with periodic multilevel oscillating boundary. Math. Methods Appl. Sci. 33, 2019–2036.CrossRefGoogle Scholar
[9]Chechkin, G. A., Friedman, A. & Piatnitski, A. L. (1999) The boundary-value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231, 213–234.Google Scholar
[10]Chechkin, G. & Piatnitski, A. L. (1999) Homogenization of boundary-value problem in a locally periodic domain. Appl. Anal. 71, 215–235.CrossRefGoogle Scholar
[11]Chechkin, G. A., Piatnitki, A. L. & Shamaev, A. S. (2007) Homogenization methods and applications. Translations of Mathematical Monographs, Vol. 234, AMS, Providence, RI.Google Scholar
[12]Cioranescu, D. & Saint, Jean Paulin J. (1998) Homogenization of reticulated structures. Applied Mathematical Sciences, Vol. 136, Springer-Verlag, Berlin, Germany.Google Scholar
[13]Dixmier, J. (1981) Von Neumann Algebras, North-Holland, Amsterdam, Netherlands.Google Scholar
[14]W. E. (2011) Principles of Multiscale Modeling, Cambridge University Press, Cambridge, UK.Google Scholar
[15]W. E. & Engquist, B. (2003) The heterogeneous multiscale methods. Comm. Math. Sci. 1, 87–132.Google Scholar
[16]Eck, C. (2004) Homogenization of a phase field model for binary mixtures. Multiscale Model. Simul. 3, 1–27.Google Scholar
[17]Ene, H. I. & Poliševski, D. (2002) Models of diffusion in partially fissured media. Z. Angew. Math. Phys. 53, 1052–1059.Google Scholar
[18]Evans, L. C. (1998) Partial differential equations. Graduate Studies in Mathematics, Vol. 19, AMS, Providence, RI.Google Scholar
[19]Fatima, T., Arab, N., Zemskov, E. P. & Muntean, A. (2011) Homogenization of a reaction-diffusion system modeling sulfate corrosion in locally periodic perforated domains. J. Eng. Math. 69, 261–276.Google Scholar
[20]Fatima, T. & Muntean, A. (to appear) Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence. Nonlinear Anal. Real World Appl.Google Scholar
[21]Fatima, T., Muntean, A. & Ptashnyk, M. (2012) Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion. Appl. Anal. 91, 1129–1154.CrossRefGoogle Scholar
[22]Gaudiello, A. (1995) Homogenization of an elliptic transmission problem. Adv. Math. Sci. Appl. 5, 639–657.Google Scholar
[23]Gilbarg, D. & Trudinger, N. S. (1983) Elliptic Partial Differential Equations of Second Order, Springer, New York.Google Scholar
[24]Gurtin, M. (1999) Configurational forces as basic concepts of continuum physics. Applied Mathematical Sciences, Springer, New York.Google Scholar
[25]Heida, M. (2011) An extension of stochastic two-scale convergence and application. Asymptotic Anal. 72, 1–30.CrossRefGoogle Scholar
[26]Le Bris, C., Legoll, F. & Thomines, F. (to appear) Rate of convergence of a two-scale expansion for some weakly stochastic homogenization problems. Asymptotic Anal.Google Scholar
[27]Lukkassen, D., Nguetseng, G. & Wall, P. (2002) Two-scale convergence. Int. J. Pure Appl. Math. 2, 35–86.Google Scholar
[28]Marchenko, V. A. & Kruslov, E. Y. (2006) Homogenization of Partial Differential Equations, Birkhäuser, Berlin, Germany.Google Scholar
[29]Marusic-Paloka, E. & Mikelic, A. (1996) An error estimate for correctors in the homogenization of the Stokes and the Navier-Stokes equations in a porous medium. Bolletino U.M.I. 10-A, 661–671.Google Scholar
[30]Meier, S. A. (2008) Two-Scale Models for Reactive Transport and Evolving Microstructure, PhD thesis, University of Bremen, Germany.CrossRefGoogle Scholar
[31]Meier, S. A. & Böhm, M. (2005) On a micro-macro system arising in diffusion-reaction problems in porous media. Proceedings of Equadiff-11, Bratislava, Slovakia, pp. 259–263.Google Scholar
[32]Meier, S. A. & Böhm, M. (2008) A note on the construction of function spaces for distributed-microstructure models with spatially varying cell geometry. Int. J. Numer. Anal. Model. (IJNAM) 1, 1–18.Google Scholar
[33]Meier, S. A. & Muntean, A. (2010) A two-scale reaction-diffusion system: Homogenization and fast-reaction limits. GAKUTO Int. Ser. Math. Sci. Appl. 32, 441–459.Google Scholar
[34]Ming, P. & Zhang, P. (2007) Analysis of the heterogeneous multi-scale method for parabolic homogenization problems. Math. Comput. 76, 153–177.CrossRefGoogle Scholar
[35]Showalter, R. E. & Walkington, J. (1991) Micro-structure models of diffusion in fissured media. J. Math. Anal. Appl. 155, 1–20.Google Scholar
[36]van Noorden, T. L. (2009) Crystal precipitation and dissolution in a thin strip. Eur. J. Appl. Math. 20, 69–91.CrossRefGoogle Scholar
[37]van Noorden, T. L. & Muntean, A. (2011) Homogenization of a locally periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493–516.Google Scholar