A stabilized finite element method for steady Darcy-Brinkman-Forchheimer flow model with different viscous and inertial resistances in porous media

We implement a stabilized finite element method for steady Darcy-Brinkman-Forchheimer model within the continuous Galerkin framework. The nonlinear fluid model is first linearized using a standard $\mathit{\text{Newton’s}}\mathit{\text{method.}}\mathit{\text{The}}\mathit{\text{sequence}}\mathit{\text{of}}\mathit{\text{linear}}\mathit{\text{problems}}\mathit{\text{is}}\mathit{\text{then}}\mathit{\text{discretized}}\mathit{\text{utilizing}}\mathit{\text{a}}\mathit{\text{stable}}\mathit{\text{inf-sup}}\mathit{\text{type}}\mathit{\text{continuous}}\mathit{\text{finite}}\mathit{\text{elements}}\mathit{\text{based}}\mathit{\text{on}}\mathit{\text{the}}\mathit{\text{Taylor-Hood}}\mathit{\text{pair}}\mathit{\text{to}}\mathit{\text{approximate}}\mathit{\text{the}}\mathit{\text{primary}}\mathit{\text{variables:}}\mathit{\text{velocity}}\mathit{\text{and}}\mathit{\text{pressure}}$. Such a pair is known to be optimal for the approximation of the isotropic Navier-Stokes equation. To overcome the well-known numerical instability in the convection-dominated problems, the Grad-Div stabilization is employed with an efficient $\mathit{\text{augmented}}\mathit{\text{Lagrangian-type}}$ penalty method. We use the penalty term to develop the $\mathit{\text{block}}\mathit{\text{Schur}}\mathit{\text{complement}}$ preconditioner, which is later coupled with a Krylov-space-based iterative linear solver. In addition, the Kelly error estimator for the adaptive mesh refinement is employed to achieve better numerical results with less computational cost. Performance of the proposed algorithm is verified for a classical benchmark problem. Particularly for the Forchheimer parameter, we present some interesting flow patterns with the velocity components and their streamlines along the mid-lines in the computational domain. The role of the Forchheimer term is highlighted for different porous medium scenarios. This study can offer an attractive setting for discretizing many multi-physics problems along with the fluid flow having inertial effects in porous media.