A Simple and Robust Weak Galerkin Method for the Brinkman Equations on Non-Convex Polytopal Meshes

Abstract

This paper presents a novel Stabilizer-Free weak Galerkin (WG) finite element method for solving the Brinkman equations without the need for conventional stabilization techniques. The Brinkman model, which mathematically blends features of both the Stokes and Darcy equations, describes fluid flow in multi-physics environments, particularly in heterogeneous porous media characterized by spatially varying permeability. In such settings, flow behavior may be governed predominantly by Darcy dynamics in certain regions and by Stokes dynamics in others. A central difficulty in this context arises from the incompatibility of standard finite element spaces: elements stable for the Stokes equations typically perform poorly for Darcy flows, and vice versa. The primary challenge addressed in this study is the development of a unified numerical scheme that maintains stability and accuracy across both flow regimes. To this end, the proposed WG method demonstrates a robust capacity to resolve both Stokes- and Darcy-dominated flows through a unified framework. The method supports general finite element partitions consisting of convex and non-convex polytopal elements, and employs bubble functions as a critical analytical component to achieve stability and convergence. Optimal-order error estimates are rigorously derived for the WG finite element solutions. Additionally, a series of numerical experiments is conducted to validate the theoretical findings, illustrating the method's robustness, reliability, flexibility, and accuracy in solving the Brinkman equations.