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

Seulip Lee, Lin Mu

TL;DR

The paper tackles robust discretization of the stationary Brinkman equations across Stokes and Darcy regimes using a minimal-enrichment enriched Galerkin (EG) framework. By applying a velocity reconstruction $\mathcal{R}$ to map EG velocities into an $H(\mathrm{div})$-conforming space and replacing the Darcy term and RHS with reconstructed velocity, the authors derive a pressure-robust PR-EG method with error estimates that are uniform in the Darcy limit ($\nu\to0$). They prove well-posedness and optimal-order convergence for both ST-EG and PR-EG, and validate the theoretical results through comprehensive 2D and 3D numerical experiments, including high-contrast permeability tests, which show clear advantages of PR-EG over ST-EG. The work offers a practical, minimal-DOF, robust solver for Stokes–Darcy type flows and suggests extensions to interface problems and nonlinear or unsteady Brinkman models.

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.

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

TL;DR

The paper tackles robust discretization of the stationary Brinkman equations across Stokes and Darcy regimes using a minimal-enrichment enriched Galerkin (EG) framework. By applying a velocity reconstruction to map EG velocities into an -conforming space and replacing the Darcy term and RHS with reconstructed velocity, the authors derive a pressure-robust PR-EG method with error estimates that are uniform in the Darcy limit (). They prove well-posedness and optimal-order convergence for both ST-EG and PR-EG, and validate the theoretical results through comprehensive 2D and 3D numerical experiments, including high-contrast permeability tests, which show clear advantages of PR-EG over ST-EG. The work offers a practical, minimal-DOF, robust solver for Stokes–Darcy type flows and suggests extensions to interface problems and nonlinear or unsteady Brinkman models.

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.
Paper Structure (19 sections, 20 theorems, 124 equations, 8 figures, 4 tables, 2 algorithms)

This paper contains 19 sections, 20 theorems, 124 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Enriched Galerkin Methods for the Brinkman equations
  4. Well-Posedness and Error Analysis for ST-EG (Algorithm \ref{['alg:EG']})
  5. Well-posedness
  6. Error estimates
  7. Well-Posedness and Error Analysis for PR-EG (Algorithm \ref{['alg:UREG']})
  8. Well-posedness
  9. Error estimates
  10. Numerical Experiments
  11. Two dimensional tests
  12. Robustness and accuracy test
  13. Error profiles with respect to $\nu$
  14. Permeability test
  15. Three dimensional tests
  16. ...and 4 more sections

Key Result

Lemma 4.1

For given $\nu$ and $h$, we define a positive constant $C_{\textsc{ne}}$ (Norm Equivalence) as where $C$ is a generic positive constant independent of $\nu$ and $h$. Then, the following norm equivalence holds: For any ${\bf v}\in{\bf V}_h$, we have for some small $0<c_1<1$. Moreover, the constant $C_{\textsc{ne}}$ is bounded as for some generic constant $C>0$.

Figures (8)

  • Figure 1: Convergence history with $h$ when different $\nu$ is given.
  • Figure 2: Numerical solutions of ST-EG and PR-EG when $\nu=10^{-6}$ and $h=1/16$.
  • Figure 3: Error profiles of the ST-EG and PR-EG methods with varying $\nu$ and a fixed mesh size $h=1/32$.
  • Figure 4: Permeability map; red regions mean $K^{-1}=10^6$ and blue regions mean $K^{-1}=1$.
  • Figure 5: Numerical velocity solutions of ST-EG and PR-EG on the permeability map.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 33 more