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

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 $\varepsilon\mathbb{Z}<sup>d$-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 $\varepsilon<sup>{n+1}$ in $L<sup>2$ and $\varepsilon<sup>n$ in $H<sup>1$, for any $n$.