Rate of convergence for correctors in almost periodic homogenization
Andriy V. Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan · Discrete and Continuous Dynamical Systems · 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 $r_n$~$1/\sqrt{n}$.