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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 and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
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