# The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE$^2$ Method

## TL;DR

This paper develops a two-scale finite element method for homogenizing linear elastic solids, extending FE-HMM to vector fields, and compares its performance and convergence with the FE$^2$ method.

## Contribution

It introduces an FE-HMM formulation for vector-valued problems in linear elasticity and provides a detailed comparison with FE$^2$, including convergence analysis and superconvergence properties.

## Key findings

- FE-HMM achieves optimal micro-macro mesh refinement strategies.
- Superconvergence properties are identified for FE-HMM.
- Numerical results show comparable accuracy with FE$^2$.

## Abstract

The Heterogeneous Multiscale Finite Element Method (FE-HMM) is a two-scale FEM based on asymptotic homogenization for solving multiscale partial differential equations. It was introduced in [W. E and B. Engquist, \emph{Commun. Math. Sci.}, 1 (2003), 87--132]. The objective of the present work is an FE-HMM formulation for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. A key ingredient of FE-HMM is that macrostiffness is estimated by stiffness sampling on heterogeneous microdomains in terms a of modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Beyond this coincidence with the Hill-Mandel macrohomogeneity condition, which is the cornerstone of the FE$^2$ method, we elaborate a conceptual comparison with the latter method. After developing an algorithmic framework we (i) assess the existing a priori convergence estimates for the micro- and macro-errors in various norms, (ii) verify optimal strategies in uniform micro-macro mesh refinements based on the estimates, (iii) analyze superconvergence properties of FE-HMM, and (iv) compare the numerical results of FE-HMM with those of FE$^2$.