Error estimate of the second-order homogenization for divergence-type nonlinear elliptic equation

TL;DR

This paper provides an error estimate for second-order homogenization of divergence-type nonlinear elliptic equations with oscillating coefficients, demonstrating that the approximation errors are locally of order O(ε) under smooth data conditions.

Contribution

It introduces a unified proof for the regularity of correctors and an O(ε) error estimate for second-order homogenization in nonlinear elliptic equations.

Findings

Error of zero-order approximation in L∞ is O(ε).

First-order approximation in Hölder norm is O(ε).

Gradient and flux errors are locally O(ε).

Abstract

Second-order two-scale expansions, a unified proof for the regularity of the correctors based on the translation invariant and a lemma for extracting $O(\epsilon)$ from the remainder term are presented for the second order nonlinear elliptic equation with rapidly oscillating coefficients. If the data are smooth enough, the error of the zero-order (or energy) in $L^\infty$, first-order in the H\"older norm, (linear periodic case)second-order's(even first-order's) gradient (or flux) in the maximum norm,are locally $O(\epsilon)$. It can be used in the parabolic equation.