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

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 method (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. We consider a MFMFE method that utilizes the lowest order Brezzi–Douglas–Marini (BDM₁) mixed finite element spaces for the approximation of velocity and pressure. The symmetric trapezoidal quadrature rule is employed for the integral of bilinear forms related to velocity variables so that the local velocity elimination is allowed which leads to a cell-centered system for pressure. We construct the multiscale space for 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 Newton iterative algorithm to solve the nonlinear problem and obtain the offline solution, which reduces the number of iterations greatly compared to the standard Picard iterative algorithm. Based on the offline basis functions and the offline solution, we calculate online basis functions on each coarse element to enrich the multiscale space iteratively. The online basis functions contain the important global information and are effective to reduce relative errors substantially. Numerical examples are provided to highlight the performance of the proposed multiscale method.