Optimal convergence rates in multiscale elliptic homogenization

Abstract

This paper is devoted to the quantitative homogenization of multiscale elliptic operator $-\nabla \cdot A_{\epsilon }\nabla$, where $A_{\epsilon }(x)=A(x/{\epsilon }_1,x/{\epsilon }_2,\cdots ,x/{\epsilon }_n)$, $\epsilon =({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\in (0,1{]}^n$ and ${\epsilon }_i>{\epsilon }_{i+1}$. We assume that $A(y_1,y_2,\cdots ,y_n)$ is 1-periodic in each $y_i\in {\mathbb{R}}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios $\max \{{\epsilon }_{i+1}/{\epsilon }_i:1\le i\le n-1\}$. In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to $\max \{e^{-c{\epsilon }_i/{\epsilon }_{i+1}}:1\le i\le n-1\}$. This convergence rate is optimal in the sense that $c>0$ cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.

71 pages