  Simulation in media with multiple continua where each continuum interacts
with every other is often challenging due to multiple scales and high contrast.
One needs some types of model reduction. One of the approaches is
multi-continuum technique, where every process in each continuum is modeled
separately and an interaction term is added. Direct numerical simulation in
multi scale media is usually not practicable. For this reason, one constructs
the corresponding homogenized equations. The paper develops a hierarchical
approach for solving cell problems at a dense network of macroscopic points
with an essentially optimal computation cost. The method employs the fact that
neighboring representative volume elements (RVEs) share similar features; and
effective properties of the neighboring RVEs are close to each other. The
hierarchical approach reduces computation cost by using different levels of
resolution for cell problems at different macroscopic points. The method
requires a hierarchy of macroscopic grid points and a corresponding nested
approximation spaces with different levels of resolution. Each level of
macroscopic points is assigned to an approximation finite element (FE) space
which is used to solve the cell problems at the macroscopic points in that
level. We prove rigorously that this hierarchical method achieves the same
level of accuracy as that of the full solve where cell problems at every
macroscopic point are solved using the FE spaces with the highest level of
resolution, but at the essentially optimal computation cost. Numerical
implementation that computes effective permeabilities of a two scale
multicontinuum system via the numerical solutions of the cell problems supports
the analytical results. Finally, we prove the homogenization convergence for
our multiscale multi-continuum system.
