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

Xiaojuan Wang, Jihong Xiao, Xiaoping Xie, Shiquan Zhang

Abstract

This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively to approximate the velocity and pressure inside the elements, and piecewise polynomials of degree $m$ to approximate their numerical traces on the interfaces of elements. In the fully discrete method, the backward Euler difference scheme is used to approximate the time derivative. The methods are shown to yield globally divergence-free velocity approximation. Optimal a priori error estimates in the energy norm and $L^2$ norm are established. A convergent linearized iterative algorithm is designed for solving the fully discrete system. Numerical experiments are provided to verify the theoretical results.

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

Abstract

This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the piecewise polynomials of degrees and respectively to approximate the velocity and pressure inside the elements, and piecewise polynomials of degree to approximate their numerical traces on the interfaces of elements. In the fully discrete method, the backward Euler difference scheme is used to approximate the time derivative. The methods are shown to yield globally divergence-free velocity approximation. Optimal a priori error estimates in the energy norm and norm are established. A convergent linearized iterative algorithm is designed for solving the fully discrete system. Numerical experiments are provided to verify the theoretical results.

Paper Structure

This paper contains 21 sections, 37 theorems, 162 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Semi-discrete WG formulations
  3. Notation and weak problem
  4. Semi-discrete WG scheme
  5. Preliminary results
  6. Well-posedness of the semi-discrete WG scheme
  7. Stability results
  8. Existence and uniqueness
  9. A priori error estimates for the semi-discrete WG scheme
  10. Primary results
  11. Error estimate
  12. Fully discrete WG method
  13. Fully discrete scheme
  14. Stability results
  15. Existence and uniqueness results
  16. ...and 6 more sections

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 8.1: Uniform triangular meshes: $4\times4$ mesh (left) and $8\times 8$ mesh (right).
  • Figure 8.2: The velocity streamlines and pressure contours for Example \ref{['EX7.3']}: $\alpha=0$ at $T=0.5$
  • Figure 8.3: The velocity streamlines and pressure contours for Example \ref{['EX7.3']}: $r=3$ and $\alpha=1,5, 50$ at $T=0.5$
  • Figure 8.4: The velocity streamlines and pressure contours for Example \ref{['EX7.2']}: $\alpha=1$ and $r=3,5,10$ at $T=0.5$
  • Figure 8.5: The domain and finite element mesh for Example \ref{['EX7.4']}
  • ...and 7 more figures

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 55 more