Weak Galerkin Methods for the Brinkman Equations

TL;DR

This paper presents a new weak Galerkin finite element method for solving the Brinkman equations, effectively handling fluid flow in heterogeneous porous media by unifying Stokes and Darcy regimes with proven stability and accuracy.

Contribution

The paper introduces a novel WG finite element method for the Brinkman equations, establishing stability and optimal error estimates, and validating with numerical experiments.

Findings

The WG method is stable and accurate for Brinkman equations.

Numerical results confirm theoretical error estimates.

The approach effectively models flow in heterogeneous porous media.

Abstract

This paper introduces a novel weak Galerkin (WG) finite element method for the numerical solution of the Brinkman equations. The Brinkman model, which seamlessly integrates characteristics of both the Stokes and Darcy equations, is employed to describe fluid flow in multiphysics contexts, particularly within heterogeneous porous media exhibiting spatially variable permeability. The proposed WG method offers a unified and robust approach capable of accurately capturing both Stokes- and Darcy-dominated regimes. A discrete inf-sup condition is established, and optimal-order error estimates are rigorously proven for the WG finite element solutions. Furthermore, a series of numerical experiments is performed to corroborate the theoretical analysis, demonstrating the method's accuracy and stability in addressing the complexities inherent in the Brinkman equations.