A uniform and pressure-robust enriched Galerkin method for the Brinkman equations

TL;DR

This paper introduces a pressure-robust enriched Galerkin method for the Brinkman equations that maintains accuracy across different flow regimes without requiring mesh size restrictions, improving stability and efficiency.

Contribution

The paper develops a novel pressure-robust EG method with a velocity reconstruction operator, ensuring uniform accuracy from Stokes to Darcy regimes while using minimal degrees of freedom.

Findings

The robust method achieves error estimates independent of pressure.

Numerical experiments confirm the theoretical stability and accuracy.

The robust method outperforms the standard approach in Darcy regimes.

Abstract

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method.