  The Boltzmann equation, as a model equation in statistical mechanics, is used
to describe the statistical behavior of a large number of particles driven by
the same physics laws. Depending on the media and the particles to be modeled,
the equation has slightly different forms. In this article, we investigate a
model Boltzmann equation with highly oscillatory media in the small Knudsen
number regime, and study the numerical behavior of the Generalized Multi-scale
Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the
media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a
general approach to numerically treat equations with multi-scale structures.
The method is divided into the offline and online steps. In the offline step,
basis functions are prepared from a snapshot space via a well-designed
generalized eigenvalue problem (GEP), and these basis functions are then
utilized to patch up for a solution through DG formulation in the online step
to incorporate specific boundary and source information. We prove the
wellposedness of the method on the Boltzmann equation, and show that the GEP
formulation provides a set of optimal basis functions that achieve spectral
convergence. Such convergence is independent of the oscillation in the media,
or the smallness of the Knudsen number, making it one of the few methods that
simultaneously achieve numerical homogenization and asymptotic preserving
properties across all scales of oscillations and the Knudsen number.

Numerical Analysis