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

TL;DR

This paper develops a stabilized finite element method for the steady Darcy-Brinkman-Forchheimer flow model, incorporating advanced stabilization, adaptive refinement, and efficient solvers to accurately simulate flow in porous media with inertial effects.

Contribution

It introduces a novel stabilized finite element approach with adaptive mesh refinement and efficient preconditioning for the Darcy-Brinkman-Forchheimer model, addressing numerical instability and computational efficiency.

Findings

Effective stabilization and adaptive refinement improve accuracy

Flow patterns vary with Forchheimer parameter in porous media

Preconditioning enhances solver performance

Abstract

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 \textit{Newton's method. The sequence of linear problems is then discretized utilizing a stable \textit{inf-sup} type continuous finite elements based on the \textit{Taylor-Hood} pair to approximate the primary variables: velocity and 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 \textit{augmented Lagrangian-type} penalty method. We use the penalty term to develop the \textit{block Schur 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.