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Robust globally divergence-free weak Galerkin methods for stationary incompressible convective Brinkman-Forchheimer equations

X. J. Wang, X. P. Xie

TL;DR

The paper develops a robust, arbitrary-order weak Galerkin discretization for the stationary incompressible convective Brinkman-Forchheimer equations, achieving globally divergence-free velocity and pressure-robust error behavior. The method uses interior velocity/pressure approximations of degrees $m$ and $m-1$ with traces on interfaces of degrees $k=m-1,m$ and stabilizes to ensure well-posedness, optimal a priori error estimates, and an $L^2$ velocity error of order $h^{m+1}$ via a duality argument. A local-elimination property reduces the global system to interface unknowns, and an Oseen-type iteration with explicit contraction conditions converges to the WG solution. Numerical experiments confirm optimal convergence rates, demonstrate divergence-free velocity, and illustrate robustness with respect to damping parameters $\alpha$ and $r$, validating both the theory and practical effectiveness.

Abstract

This paper develops a class of robust weak Galerkin methods for the stationary incompressible convective Brinkman-Forchheimer equations. The methods adopt piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively for the approximations of velocity and pressure variables inside the elements and piecewise polynomials of degrees $k \ ( k=m-1,m)$ and $m$ respectively for their numerical traces on the interfaces of elements, and are shown to yield globally divergence-free velocity approximation. Existence and uniqueness results for the discrete schemes, as well as optimal a priori error estimates, are established. A convergent linearized iterative algorithm is also presented. Numerical experiments are provided to verify the performance of the proposed methods

Robust globally divergence-free weak Galerkin methods for stationary incompressible convective Brinkman-Forchheimer equations

TL;DR

The paper develops a robust, arbitrary-order weak Galerkin discretization for the stationary incompressible convective Brinkman-Forchheimer equations, achieving globally divergence-free velocity and pressure-robust error behavior. The method uses interior velocity/pressure approximations of degrees and with traces on interfaces of degrees and stabilizes to ensure well-posedness, optimal a priori error estimates, and an velocity error of order via a duality argument. A local-elimination property reduces the global system to interface unknowns, and an Oseen-type iteration with explicit contraction conditions converges to the WG solution. Numerical experiments confirm optimal convergence rates, demonstrate divergence-free velocity, and illustrate robustness with respect to damping parameters and , validating both the theory and practical effectiveness.

Abstract

This paper develops a class of robust weak Galerkin methods for the stationary incompressible convective Brinkman-Forchheimer equations. The methods adopt piecewise polynomials of degrees and respectively for the approximations of velocity and pressure variables inside the elements and piecewise polynomials of degrees and respectively for their numerical traces on the interfaces of elements, and are shown to yield globally divergence-free velocity approximation. Existence and uniqueness results for the discrete schemes, as well as optimal a priori error estimates, are established. A convergent linearized iterative algorithm is also presented. Numerical experiments are provided to verify the performance of the proposed methods
Paper Structure (13 sections, 24 theorems, 121 equations, 12 figures, 2 tables)

This paper contains 13 sections, 24 theorems, 121 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Weak Galerkin finite element scheme
  3. Notation and weak problem
  4. WG scheme
  5. Preliminary results
  6. Well-posedness of discrete scheme
  7. A priori error estimates
  8. $L^2$ error estimation for velocity
  9. Local elimination property and iteration scheme
  10. Local elimination property
  11. Iteration scheme
  12. Numerical experiments
  13. Conclusion

Key Result

Theorem 2.1

Let $\bm{u}_h=\{\bm{u}_{hi},\bm{u}_{hb} \}\in \bm{V}_h^{0}$ be the velocity solution of the WG scheme WG. Then there hold

Figures (12)

  • Figure 7.1: Uniform triangular meshes: $4\times4$ mesh (left) and $8\times 8$ mesh (right).
  • Figure 7.2: The velocity streamlines and pressure contours for Example \ref{['EX7.2']}: $\alpha=0$
  • Figure 7.3: The velocity streamlines and pressure contours for Example \ref{['EX7.2']}: $r=5$ and $\alpha=1, 50, 100$
  • Figure 7.4: The velocity streamlines and pressure contours for Example \ref{['EX7.2']}: $\alpha=5$ and $r=3,5,50$
  • Figure 7.5: The domain and finite element mesh for Example \ref{['EX7.3']}
  • ...and 7 more figures

Theorems & Definitions (42)

  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • Lemma 2.2: cf. bookFEM
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 32 more