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 non-periodic 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. We assume that the local solutions can be represented as a combination of a linear interpolation of local solutions 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 $\mathcal{O}(1)$, we establish that our approach incurs a cost of $\mathcal{O}(L \eta^{(1-L)d})$, highlighting substantial computational savings across hierarchical layers $L$, coarsening factor $\eta$, and spatial dimension $d$. Numerical experiments validate the effectiveness of the proposed method in media with slowly varying properties, underscoring its potential for efficient multiscale modeling.

翻译：近年来发展的一种上尺度化技术——多连续介质均匀化方法，因其在复杂多尺度系统建模中的有效性而受到广泛关注。该方法基于不同的物理性质定义多个连续介质，并通过求解一系列约束单元问题来捕获每个连续介质的局部信息。然而，在非周期性问题中，大多数均匀化方法普遍存在一个局限性：在每个宏观点上使用极细网格求解所有这些单元问题计算成本高昂。为应对这一挑战，本文提出了一种分层多连续介质均匀化框架。其核心思想是定义分层宏观点，并在不同分辨率的网格上求解约束问题。我们假设局部解可以表示为前一层级局部解的线性插值与一个附加修正项的组合。将此组合代入原始约束问题后，修正项将根据宏观点所在层级，使用不同尺寸的有限元网格进行求解。通过将完全求解局部问题的计算成本归一化为 $\mathcal{O}(1)$，我们证明所提方法的计算成本为 $\mathcal{O}(L \eta^{(1-L)d})$，这显著体现了该方法在分层层级 $L$、粗化因子 $\eta$ 和空间维度 $d$ 上带来的巨大计算节约。数值实验验证了该方法在性质缓慢变化的介质中的有效性，凸显了其在高效多尺度建模方面的潜力。