  In this paper, we propose a multiscale method for the Darcy-Forchheimer model
in highly heterogeneous porous media. The problem is solved in the framework of
generalized multiscale finite element methods (GMsFEM) combined with a
multipoint flux mixed finite element (MFMFE) method. %In the MFMFE methods,
appropriate mixed finite element spaces and suitable quadrature rules are
employed, which allow for local velocity elimination and lead to a
cell-centered system for the pressure. We consider the MFMFE method that
utilizes the lowest order Brezzi-Douglas-Marini ($\textrm{BDM}_1$) mixed finite
element spaces for the velocity and pressure approximation. The symmetric
trapezoidal quadrature rule is employed for the integration of bilinear forms
relating to the velocity variables so that the local velocity elimination is
allowed and leads to a cell-centered system for the pressure. %on meshes
composed of simplices and $h^2$-perturbed parallelograms. We construct
multiscale space for the pressure and solve the problem on the coarse grid
following the GMsFEM framework. In the offline stage, we construct local
snapshot spaces and perform spectral decompositions to get the offline space
with a smaller dimension. In the online stage, we use the Newton iterative
algorithm to solve the nonlinear problem and obtain the offline solution, which
reduces the iteration times greatly comparing to the standard Picard iteration.
Based on the offline space and offline solution, we calculate online basis
functions which contain important global information to enrich the multiscale
space iteratively. The online basis functions are efficient and accurate to
reduce relative errors substantially. Numerical examples are provided to
highlight the performance of the proposed multiscale method.
