Computation of quasilocal effective diffusion tensors and connections to the mathematical theory of homogenization
For researchers in multiscale modeling and homogenization, this work provides a theoretical connection between numerical and analytical approaches, but the results are largely theoretical and incremental.
This paper bridges numerical and analytical homogenization by reinterpreting a multiscale method via a discrete integral operator, proving that a localized piecewise constant coefficient coincides with the classical homogenization limit in periodic settings and providing error analysis for non-periodic cases.
This paper aims at bridging existing theories in numerical and analytical homogenization. For this purpose the multiscale method of Målqvist 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.