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Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

Author

Listed:
  • Xiaoqin Guo

    (University of Wisconsin Madison)

  • Hung V. Tran

    (University of Wisconsin Madison)

  • Yifeng Yu

    (University of California)

Abstract

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $$O(\varepsilon )$$ O ( ε ) or $$O(\varepsilon ^2)$$ O ( ε 2 ) depending on the diffusion matrix A, source term f, and boundary data g. Moreover, we show that the set of diffusion matrices A that give optimal rate $$O(\varepsilon )$$ O ( ε ) is open and dense in the set of $$C^2$$ C 2 periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $$O(\varepsilon )$$ O ( ε ) .

Suggested Citation

  • Xiaoqin Guo & Hung V. Tran & Yifeng Yu, 2020. "Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form," Partial Differential Equations and Applications, Springer, vol. 1(4), pages 1-16, August.
    • Handle: RePEc:spr:pardea:v:1:y:2020:i:4:d:10.1007_s42985-020-00017-z
    DOI: 10.1007/s42985-020-00017-z
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