  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.
