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In this paper, we propose a multiscale method for the Darcy-Forchheimer modelin highly heterogeneous porous media. The problem is solved in the framework ofgeneralized multiscale finite element methods (GMsFEM) combined with amultipoint flux mixed finite element (MFMFE) method. %In the MFMFE methods,appropriate mixed finite element spaces and suitable quadrature rules areemployed, which allow for local velocity elimination and lead to acell-centered system for the pressure. We consider the MFMFE method thatutilizes the lowest order Brezzi-Douglas-Marini ($\textrm{BDM}_1$) mixed finiteelement spaces for the velocity and pressure approximation. The symmetrictrapezoidal quadrature rule is employed for the integration of bilinear formsrelating to the velocity variables so that the local velocity elimination isallowed and leads to a cell-centered system for the pressure. %on meshescomposed of simplices and $h^2$-perturbed parallelograms. We constructmultiscale space for the pressure and solve the problem on the coarse gridfollowing the GMsFEM framework. In the offline stage, we construct localsnapshot spaces and perform spectral decompositions to get the offline spacewith a smaller dimension. In the online stage, we use the Newton iterativealgorithm to solve the nonlinear problem and obtain the offline solution, whichreduces the iteration times greatly comparing to the standard Picard iteration.Based on the offline space and offline solution, we calculate online basisfunctions which contain important global information to enrich the multiscalespace iteratively. The online basis functions are efficient and accurate toreduce relative errors substantially. Numerical examples are provided tohighlight the performance of the proposed multiscale method.

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model