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

2020/01/06 by Xiaoqin Guo, Guo, Xiaoqin, Hung V. Tran +3 · 2 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2001.01712

openalex publication_date 2020/01/06 · openalex created_date 2020/01/10 · openalex updated_date 2026/07/28

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(ε) or O(ε²) 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(ε) is open and dense in the set of C^2,α periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is O(ε).

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• Homogenization of non-divergence form operators in i.i.d. random environments
• Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface

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