Higher-order homogenization for equations of linearized elasticity using the operator-asymptotic approach

TL;DR

This paper extends the operator-asymptotic homogenization approach to three-dimensional linearized elasticity, achieving higher-order convergence rates for periodic composite media using Bloch approximation.

Contribution

It introduces a higher-order homogenization method for linearized elasticity using the operator-asymptotic approach with Bloch approximation, providing improved convergence rates.

Findings

Achieves error of order ε^{n+1} in L^2 norm

Achieves error of order ε^n in H^1 norm

Extends the approach to vector-valued functions with compact Fourier support

Abstract

The operator-asymptotic approach was introduced by Lim-\v{Z}ubrini\'c in [Asymptotic Analysis. 141(4), p. 211-256 (2025)] for the homogenization of an $\varepsilon\mathbb{Z}^d$-periodic composite media. In this article, we consider the setting of three-dimensional linearized elasticity, and extend the approach to obtain higher-order convergence rates. In particular, we consider the so-called ``Bloch approximation'' for vector-valued functions with compact Fourier support, and demonstrate that under such data, the approach provides an expansion that yields an error of order $\varepsilon^{n+1}$ in $L^2$ and $\varepsilon^n$ in $H^1$, for any $n$.