Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed virtual element approximation for the five-field formulation of the steady Boussinesq problem with temperature-dependent parameters

Zeinab Gharibi

Abstract

In this work, we develop recent research on the fully mixed virtual element method (mixed-VEM) based on the Banach space for the stationary Boussinesq equation to suggest and analyze a new mixed-VEM for the stationary two-dimensional Boussinesq equation with temperature-dependent parameters in terms of the pseudostress, vorticity, velocity, pseudoheat vector and temperature fields. The well-posedness of the continuous formulation is analyzed utilizing a fixed-point strategy, a smallness assumption on the data, and some additional regularities on the solution. The discretization for the mentioned variables is based on the coupling $\mathbb{H}(\mathbf{div}_{6/5})$ -- and $\mathbf{H}(\mathrm{div}_{6/5})$ -- conforming virtual element techniques. The proposed scheme is rewritten as an equivalent fixed point operator equation, so that its existence and stability estimates have been proven. In addition, an a priori convergence analysis is established by utilizing the Céa estimate and a suitable assumption on data for all variables in their natural norms showing an optimal rate of convergence. Finally, several numerical examples are presented to illustrate the performance of the proposed method.

Mixed virtual element approximation for the five-field formulation of the steady Boussinesq problem with temperature-dependent parameters

Abstract

In this work, we develop recent research on the fully mixed virtual element method (mixed-VEM) based on the Banach space for the stationary Boussinesq equation to suggest and analyze a new mixed-VEM for the stationary two-dimensional Boussinesq equation with temperature-dependent parameters in terms of the pseudostress, vorticity, velocity, pseudoheat vector and temperature fields. The well-posedness of the continuous formulation is analyzed utilizing a fixed-point strategy, a smallness assumption on the data, and some additional regularities on the solution. The discretization for the mentioned variables is based on the coupling -- and -- conforming virtual element techniques. The proposed scheme is rewritten as an equivalent fixed point operator equation, so that its existence and stability estimates have been proven. In addition, an a priori convergence analysis is established by utilizing the Céa estimate and a suitable assumption on data for all variables in their natural norms showing an optimal rate of convergence. Finally, several numerical examples are presented to illustrate the performance of the proposed method.
Paper Structure (27 sections, 22 theorems, 254 equations, 4 figures)

This paper contains 27 sections, 22 theorems, 254 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. The continuous formulation
  4. The model problem
  5. The variational formulation
  6. The continuous solvability analysis
  7. Preliminaries
  8. A fixed point strategy
  9. Well-posedness of the uncoupled problems
  10. Solvability analysis of the fixed-point equation
  11. Mixed virtual element approximation
  12. Preliminaries
  13. Discrete spaces
  14. Discrete virtual forms and the discrete problem
  15. Solvability Analysis: discrete scheme
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $X$ and $Y$ be reflexive Banach spaces, and let $a: X\times X \rightarrow\mathrm{R}$ and $b: X\times Q \rightarrow\mathrm{R}$ be given bounded bilinear forms. Moreover, let $\mathbf{B}: X\rightarrow Y'$ be the bounded linear operator induced by $b$, and let $\mathcal{H}:=N(\mathbf{B})$ be the re Then, for each pair $(f , g)\in X' \times Y'$ there exists a unique $(u,p)\in X\times Y$ such that

Figures (4)

  • Figure 7.1: Example 1, illustration of the meshes used: non-convex mesh (left panel), hexagon mesh (central panel) and quadrilateral mesh (right panel).
  • Figure 7.2: Example 1, snapshots of the first and fourth components of numerical stress, and velocity magnitude (first row, left to right), vorticity magnitude, pressure and pseudoheat magnitude (second row, left to right), and temperature (third row), computed with $r = 0$ in the mesh made of distortion quadrilaterals with $h = 1/32$.
  • Figure 7.3: Example 2, snapshots of the stress, velocity, and vorticity magnitudes (first row, left to right) and the pressure and heat-flux vector magnitudes, and temperature (second row, left to right) computed with $r = 0$ in a mesh made of polygonals with $h = 3.030$e-2.
  • Figure 7.4: Example 3, snapshots of stress, first component of velocity, vorticity and temperature for values $\mathrm{Ra}=1$e2, computed with $r = 0$ in a polgonal mesh with $h = 1.39$e-2.

Theorems & Definitions (40)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 30 more