**Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model**

#### **Denis Spiridonov 1, Jian Huang 2,3,4, Maria Vasilyeva 5,6,\*, Yunqing Huang 2,3,4 and Eric T. Chung 7**


Received: 7 October 2019; Accepted: 5 December 2019; Published: 10 December 2019

**Abstract:** In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.

**Keywords:** Darcy-Forchheimer model; flow in porous media; nonlinear equation; heterogeneous media; finite element method; multiscale method; mixed generalized multiscale finite element method; multiscale basis functions; two-dimensional domain
