American Institute of Mathematical Sciences

Advanced
February  2005, 13(2): 503-514. doi: 10.3934/dcds.2005.13.503

Rate of convergence for correctors in almost periodic homogenization

Andriy Bondarenko 1, , Guy Bouchitté 2, , Luísa Mascarenhas 3, and  Rajesh Mahadevan 4,

1. 

Centre de Physique Théorique-CNRS, Luminy Case 907, 13288 Marseille Cedex 09, France
2. 

Département de Mathématiques, Université de Toulon et du Var-BP 20132, 83957 La Garde Cedex, France
3. 

FCT/UNL and CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa Codex, Portugal
4. 

CMM, Universidad de Chile, Av. Blanco Encalada 2120, Casilla 170/3, Santiago, Chile

Received  April 2004 Revised  January 2005 Published  April 2005

In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the zero order corrector $u_n -u^{h o m}$ in the $L^2$ norm is $1/n$, the same as the scale of periodicity (see Jikov et al [6]). It is possible to have the same rate of convergence in the case of almost periodic coefficients under some stringent structural conditions on the coefficients (see Kozlov [7]). The goal of this note is to construct almost periodic media where the rate of convergence is lower than $1/n$. To that aim, in the one dimensional setting, we introduce a family of random almost periodic coefficients for which we compute, using Fourier series analysis, the mean rate of convergence $r_n$ (mean with respect to the random parameter). This allows us to present examples where we find $r_n$>>$1/n^r$ for every $r>0$, showing a big contrast with the random case considered by Bourgeat and Piatnitski [2] where $r_n$~$1/\sqrt{n}$.
Keywords: irrationality measure., Homogenization, correctors, rate of convergence, Fourier analysis, random almost periodic coefficients.
Mathematics Subject Classification: 35B27, 35B40, 42A75, 11J82.
Citation: Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[1]

Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks & Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97
[2]

Zhanying Yang. Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method. Communications on Pure & Applied Analysis, 2014, 13 (1) : 249-272. doi: 10.3934/cpaa.2014.13.249
[3]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[4]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745
[5]

Emmanuel Gobet, Mohamed Mrad. Convergence rate of strong approximations of compound random maps, application to SPDEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4455-4476. doi: 10.3934/dcdsb.2018171
[6]

Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks & Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
[7]

Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-9. doi: 10.3934/dcdss.2020087
[8]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49
[9]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[10]

Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
[11]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[12]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[13]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[14]

Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021
[15]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[16]

Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks & Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523
[17]

Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[18]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47
[19]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
[20]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences