Stabilized equal lower-order finite element methods for simulating Brinkman equations in porous media

Abstract

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the $L^2$ norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations.

Keywords:

• Galerkin least-squares
• least-squares
• equal-order finite elements
• stabilization finite element method
• pore-scale flow

AMS Subject Classifications:

• 76-10
• 76M10
• 65N30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author gratefully acknowledges the financial support provided in part by the National Science and Technology Council of Taiwan under grant 112-2115-M-160-001. The second author is grateful for the financial support provided in part by the US National Science Foundation under grant DMS-2207971.