Computation of quasilocal effective diffusion tensors and connections to the mathematical theory of homogenization

TL;DR

This paper develops a localized numerical method for computing effective diffusion tensors in homogenization, connecting multiscale finite element techniques with classical theory, and validates it through analysis and numerical experiments.

Contribution

It reinterprets a multiscale method via integral operators, introduces a localized coefficient approximation, and proves its consistency with classical homogenization in periodic settings.

Findings

Localized coefficients match classical homogenization in periodic cases

The method provides accurate approximations beyond periodic settings

Numerical experiments confirm theoretical predictions

Abstract

This paper aims at bridging existing theories in numerical and analytical homogenization. For this purpose the multiscale method of M{\aa}lqvist and Peterseim [Math. Comp. 2014], which is based on orthogonal subspace decomposition, is reinterpreted by means of a discrete integral operator acting on standard finite element spaces. The exponential decay of the involved integral kernel motivates the use of a diagonal approximation and, hence, a localized piecewise constant coefficient. In a periodic setting, the computed localized coefficient is proved to coincide with the classical homogenization limit. An a priori error analysis shows that the local numerical model is appropriate beyond the periodic setting when the localized coefficient satisfies a certain homogenization criterion, which can be verified a posteriori. The results are illustrated in numerical experiments.