A Hierarchical Approach for Multicontinuum Homogenization in High Contrast Media
Wei Xie, Viêt Hà Hòang, Yin Yang, Yunqing Huang · SIAM Journal on Scientific Computing · 2026
Abstract. A recently developed upscaling technique, the multicontinuum homogenization method, has gained significant attention for its effectiveness in modeling complex multiscale systems. This method defines multiple continua based on distinct physical properties and solves a series of constrained cell problems to capture localized information for each continuum. However, solving all these cell problems on very fine grids at every macroscopic point is computationally expensive, which is a common limitation of most homogenization approaches for nonperiodic problems. To address this challenge, we propose a hierarchical multicontinuum homogenization framework. The core idea is to define hierarchical macroscopic points and solve the constrained problems on grids of varying resolutions. The local solutions are decomposed into the linear interpolation of contributions inherited from preceding levels and an additional correction term. This combination is substituted into the original constrained problems, and the correction term is resolved using finite element (FE) grids of varying sizes depending on the level of the macropoint. By normalizing the computational cost of fully resolving the local problem to [Formula: see text], we establish that our approach incurs a cost of [Formula: see text], highlighting substantial computational savings across hierarchical layers [Formula: see text], coarsening factor [Formula: see text], and spatial dimension [Formula: see text]. Numerical experiments validate the effectiveness of the proposed method in media with slowly varying properties, underscoring its potential for efficient multiscale modeling. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/xieweidc/hierarchical_mh_elliptic.git and in the supplementary materials ( hierarchical_mh_elliptic-main.zip [19.8KB]). [Formula: see text]