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A virtual element method for a convective Brinkman-Forchheimer problem coupled with a heat equation

Danilo Amigo, Felipe Lepe, Enrique Otarola, Gonzalo Rivera

TL;DR

This work develops a divergence-free virtual element method for a nonlinear, temperature-coupled flow modeled by the convective Brinkman--Forchheimer equations with temperature-dependent viscosity and diffusion. The method uses polygonal meshes to approximate velocity, pressure, and temperature while preserving a discrete divergence-free kernel, and it provides existence, stability, and optimal error estimates via a fixed-point framework and AVV-inspired constructions. Extending previous AVV results, the authors incorporate a nonlinear Forchheimer term and a temperature-dependent diffusion coefficient, and they validate the theory with numerical experiments across multiple mesh families, including a viscosity-robust exploration. Overall, the paper contributes a rigorous, flexible discretization tool for nonlinear, non-isothermal flows on general meshes, with implications for accurate, geometry-agnostic simulations in complex geometries.

Abstract

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under appropriate regularity assumptions for the solution. We conclude with a series of numerical tests performed with different mesh families that complement our theoretical findings.

A virtual element method for a convective Brinkman-Forchheimer problem coupled with a heat equation

TL;DR

This work develops a divergence-free virtual element method for a nonlinear, temperature-coupled flow modeled by the convective Brinkman--Forchheimer equations with temperature-dependent viscosity and diffusion. The method uses polygonal meshes to approximate velocity, pressure, and temperature while preserving a discrete divergence-free kernel, and it provides existence, stability, and optimal error estimates via a fixed-point framework and AVV-inspired constructions. Extending previous AVV results, the authors incorporate a nonlinear Forchheimer term and a temperature-dependent diffusion coefficient, and they validate the theory with numerical experiments across multiple mesh families, including a viscosity-robust exploration. Overall, the paper contributes a rigorous, flexible discretization tool for nonlinear, non-isothermal flows on general meshes, with implications for accurate, geometry-agnostic simulations in complex geometries.

Abstract

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under appropriate regularity assumptions for the solution. We conclude with a series of numerical tests performed with different mesh families that complement our theoretical findings.
Paper Structure (28 sections, 10 theorems, 118 equations, 7 figures, 1 table)

This paper contains 28 sections, 10 theorems, 118 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Notation and preliminary remarks
  5. Notation
  6. Data assumptions
  7. A convective Brinkman--Forchheimer problem
  8. Existence, stability, and uniqueness results
  9. A nonlinear heat equation
  10. Existence, stability, and uniqueness results
  11. The coupled problem
  12. Existence, stability, and uniqueness results
  13. A virtual element approximation
  14. Mesh regularity
  15. Basic ingredients
  16. ...and 13 more sections

Key Result

Proposition 1

\newlabelpro:existencia10 Let $\Omega\subset \mathbb{R}^2$ be an open and bounded domain with Lipschitz boundary $\partial\Omega$, and let $\boldsymbol{f}\in[\mathrm{H}^{-1}(\Omega)]^2$. Then, there exists at least one solution $(\boldsymbol{u},\textsf{p})\in\mathbf{V}\times\mathrm{Q}$ for problem where $\Lambda(\boldsymbol{f}) := 1 + \nu^*\nu_*^{-1} + C_N\nu_*^{-2}\|\boldsymbol{f}\|_{-1}+\textsf

Figures (7)

  • Figure 1: Initial meshes. From top left to bottom right: ${\mathcal{T}}_h^1$, ${\mathcal{T}}_h^2$, ${\mathcal{T}}_h^3$, ${\mathcal{T}}_h^4$, ${\mathcal{T}}_h^5$, and ${\mathcal{T}}_h^6$, with $N=8$.
  • Figure 2: Test 1: Experimental convergence rates for $|e_{\boldsymbol{u}}|_{1,\Omega}$, $|e_{T}|_{1,\Omega}$, and $\|e_{p}\|_{0,\Omega}$ on different polygonal meshes.
  • Figure 3: Test 2: Experimental convergence rates for $|e_{\boldsymbol{u}}|_{1,\Omega}$, $|e_{T}|_{1,\Omega}$, and $\|e_{p}\|_{0,\Omega}$ on different polygonal meshes.
  • Figure 4: Virtual element approximatiopn: magnitude of the velocity $\boldsymbol{u}_h$ (left above), pressure $p_h$ (right above), and temperature $T_h$ (below) in ${\mathcal{T}}_h^5$
  • Figure 5: Test 3: Velocity error for $\nu =10^{-1},10^{-4},10^{-8}$ on the meshes ${\mathcal{T}}_h^1, {\mathcal{T}}_h^2$, and ${\mathcal{T}}_h^3$.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Proposition 1: existence, stability, and uniqueness
  • Proposition 2: existence, stability, and uniqueness
  • Remark 3: skew-symmetry
  • Proposition 4: existence, stability, and uniqueness
  • Remark 5: computability
  • Remark 6: divergence-free
  • Remark 7: computability
  • Remark 8: skew-symmetry
  • Remark 9: the no discretization of $b$
  • Remark 10: reduced formulation
  • ...and 14 more