Rate of convergence for correctors in almost periodic homogenization

Andriy Bondarenko, Guy Bouchitté, Luisa Mascarenhas, Rajesh Mahadevan

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

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 un - uhom in the L2 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 rn (mean with respect to the random parameter). This allows us to present examples where we find rn ≫ 1/nr for every r > 0, showing a big contrast with the random case considered by Bourgeat and Piatnitski [2] where rn ∼ 1/√n.

Original languageEnglish
Pages (from-to)503-514
Number of pages12
JournalDiscrete And Continuous Dynamical Systems
Volume13
Issue number2
DOIs
Publication statusPublished - Jul 2005

Keywords

  • Correctors
  • Fourier analysis
  • Homogenization
  • Irrationality measure
  • Random almost periodic coefficients
  • Rate of convergence

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