In this paper, we provide an analysis of a recently proposed multicontinuum homogenization technique. The analysis differs from those used in classical homogenization methods for several reasons. First, the cell problems in multicontinuum homogenization use constraint problems and can not be directly substituted into the differential operator. Secondly, the problem contains high contrast that remains in the homogenized problem. The homogenized problem averages the microstructure while containing the small parameter. In this analysis, we first based on our previous techniques, CEM-GMsFEM, to define a CEM-downscaling operator that maps the multicontinuum quantities to an approximated microscopic solution. Following the regularity assumption of the multicontinuum quantities, we construct a downscaling operator and the homogenized multicontinuum equations using the information of linear approximation of the multicontinuum quantities. The error analysis is given by the residual estimate of the homogenized equations and the well-posedness assumption of the homogenized equations.

翻译：本文对近期提出的多连续介质均匀化技术进行了分析。该分析方法与经典均匀化方法存在若干差异：首先，多连续介质均匀化中的胞元问题采用约束问题，无法直接代入微分算子；其次，问题本身包含的高对比度特性会保留在均匀化问题中。该均匀化问题在包含小参数的同时对微观结构进行平均处理。本分析首先基于我们先前提出的CEM-GMsFEM技术，定义了将多连续介质量映射至近似微观解的CEM降尺度算子。基于多连续介质量的正则性假设，我们利用其线性近似信息构建了降尺度算子及均匀化多连续介质方程。误差分析通过均匀化方程的残差估计及其适定性假设完成。