Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations

TL;DR

This paper establishes optimal convergence rates for elliptic homogenization problems in nondivergence form, providing theoretical analysis and numerical validation of the rates in various norms.

Contribution

It derives the optimal convergence rate of (\u03b5) in the W^{1,p} norm and supplies explicit bounds with correction terms, supported by numerical experiments.

Findings

Optimal () convergence rate in W^{1,p} norm.

Explicit gradient and Hessian bounds with correction terms.

Numerical experiments confirm theoretical optimality.

Abstract

We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of $u^{\varepsilon}$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\varepsilon)$. We further obtain optimal gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm. We then provide an explicit $c$-bad diffusion matrix and use it to perform various numerical experiments, which demonstrate the optimality of the obtained rates.