The Well-Posedness and Discontinuous Galerkin Approximation for the Non-Newtonian Stokes–Darcy–Forchheimer Coupling System

We study the non-Newtonian Stokes–Darcy–Forchheimer system modeling the free fluid coupled with the porous medium flow with shear/velocity-dependent viscosities. The unique existence is proved by using the theory of nonlinear monotone operator and a coupled inf-sup condition. Moreover, we apply the discontinuous Galerkin (DG) method with Pk/Pk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^k/P^{k-1}$$\end{document}-DG element for numerical discretization and obtain the well-posedness, stability, and error estimate. For both the continuous and the discrete problem, we explore the convergence of the Picard iteration (or called Kacǎnov method). The theoretical results are confirmed by the numerical examples.