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：本文对近期提出的一种多连续介质均匀化技术进行了分析。该分析与经典均匀化方法中的分析存在若干差异。首先，多连续介质均匀化中的胞元问题使用了约束问题，无法直接代入微分算子。其次，问题中含有保留在均匀化问题中的高对比度特征。均匀化问题在包含小参数的同时对微观结构进行了平均。在本分析中，我们首先基于先前提出的CEM-GMsFEM技术，定义了一个CEM降尺度算子，该算子将多连续介质量映射到近似的微观解。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均匀化方程的适定性假设完成。