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

Abstract

The operator-asymptotic approach was introduced by Lim-Žubrinić in [Asymptotic Analysis. 141(4), p. 211-256 (2025)] for the homogenization of an $\epsilon {\mathbb{Z}}^d$-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 ${\epsilon }^{n+1}$ in $L^2$ and ${\epsilon }^n$ in $H^1$, for any $n$.

35 pages, 2 figures, 1 table