Rate of convergence for correctors in almost periodic homogenization

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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

Abstract
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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

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