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

Xiaoqin Guo (), Hung V. Tran () and Yifeng Yu ()
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Xiaoqin Guo: University of Wisconsin Madison
Hung V. Tran: University of Wisconsin Madison
Yifeng Yu: University of California

Partial Differential Equations and Applications, 2020, vol. 1, issue 4, 1-16

Abstract: 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 ) .

Keywords: Homogenization; Periodic setting; Linear non-divergence form elliptic equations; Optimal rates of convergence; 35B27; 35B40; 35D40; 35J25; 49L25 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00017-z

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