Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A priori and a posteriori error estimates of a really pressure-robust virtual element method for the incompressible Brinkman problem

Yu Xiong, Yanping Chen

TL;DR

This work develops a really pressure-robust virtual element method for the incompressible Brinkman equations on polygonal meshes, leveraging a divergence-preserving Raviart-Thomas reconstruction to discretize the RHS. The authors prove optimal a priori error estimates with velocity errors independent of the continuous pressure and viscosity, and they devise a residual-based a posteriori error estimator with global reliability and local efficiency, enabling adaptive mesh refinement. The methodology is supported by extensive numerical experiments demonstrating robustness across high-contrast permeability, polygonal meshes, and geometric singularities, with substantial improvement over standard VEM in avoiding locking as $\nu \to 0$. Collectively, the results provide a practical, reliable framework for simulating Brinkman-type flows in complex porous media with efficient adaptivity.

Abstract

This paper presents both a priori and a posteriori error analyses for a really pressure-robust virtual element method to approximate the incompressible Brinkman problem. We construct a divergence-preserving reconstruction operator using the Raviart-Thomas element for the discretization on the right-hand side. The optimal priori error estimates are carried out, which imply the velocity error in the energy norm is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method's ability to handle more general polygonal meshes, we implement effective mesh refinement strategies and develop a residual-type a posteriori error estimator. This estimator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method.

A priori and a posteriori error estimates of a really pressure-robust virtual element method for the incompressible Brinkman problem

TL;DR

This work develops a really pressure-robust virtual element method for the incompressible Brinkman equations on polygonal meshes, leveraging a divergence-preserving Raviart-Thomas reconstruction to discretize the RHS. The authors prove optimal a priori error estimates with velocity errors independent of the continuous pressure and viscosity, and they devise a residual-based a posteriori error estimator with global reliability and local efficiency, enabling adaptive mesh refinement. The methodology is supported by extensive numerical experiments demonstrating robustness across high-contrast permeability, polygonal meshes, and geometric singularities, with substantial improvement over standard VEM in avoiding locking as . Collectively, the results provide a practical, reliable framework for simulating Brinkman-type flows in complex porous media with efficient adaptivity.

Abstract

This paper presents both a priori and a posteriori error analyses for a really pressure-robust virtual element method to approximate the incompressible Brinkman problem. We construct a divergence-preserving reconstruction operator using the Raviart-Thomas element for the discretization on the right-hand side. The optimal priori error estimates are carried out, which imply the velocity error in the energy norm is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method's ability to handle more general polygonal meshes, we implement effective mesh refinement strategies and develop a residual-type a posteriori error estimator. This estimator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method.

Paper Structure

This paper contains 17 sections, 9 theorems, 101 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The continuous problem
  4. The Helmholtz decomposition
  5. A really pressure-robust Virtual element
  6. Projection operators and virtual element spaces
  7. Divergence-preserving reconstruction operator
  8. Discrete scheme
  9. A priori error analysis
  10. A posteriori error analysis and mesh adaptivity
  11. Upper bound
  12. Local Lower Bound
  13. An adaptive mesh refinement
  14. Numerical results
  15. Numerical tests for a priori error estimate
  16. ...and 2 more sections

Key Result

Lemma 3.1

For all $\mathbf{v}\in \mathbf{V}$, the reconstruction operator has the following properties where $C_{\mathcal{R}_1}$ is a positive constant independent of $h$.

Figures (11)

  • Figure 3.1: Illustration of degrees of freedom. We represent $\mathbf{D}_{\mathbf{V}}1$ with the blue dots, $\mathbf{D}_{\mathbf{V}}2$ with the red arrows.
  • Figure 3.2: A subtriangulation of a pentagon $E$ and the corresponding notations.
  • Figure 5.1: Illustration of refining a local heptagon.
  • Figure 6.1: Illustration of three types of meshes.
  • Figure 6.2: Example 6.2: Numerical results. (a) Profile of $\kappa^{-1}$ for fibrous structure; (b) Velocity field of $\mathbf{u}_h$; (c) First component of velocity $u_{h,1}$; (d) second component of velocity $u_{h,2}$; (e) pressure profile $p_h$;
  • ...and 6 more figures

Theorems & Definitions (24)

  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.1
  • proof
  • Remark 4.1
  • Corollary 4.1
  • ...and 14 more